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METHODS IN CLIMATOLOGY 



LONDON: GEOFFREY CUMBERLEGE 
OXFORD UNIVERSITY PRESS 



METHODS IN 

CLIMATOLOGY 



BY 

VICTOR CONRAD 

HARVARD UNIVERSITY 




CAMBRIDGE, MASSACHUSETTS 

HARVARD UNIVERSITY PRESS 
1946 



COPYRIGHT, 1944 

BY THE PRESIDENT AND FELLOWS OF HARVARD COLLEGE 
Second Printing 



Printed in the United States of America by 
Lancaster Press, Inc., Lancaster, Penna. 



TO 

IDA CONRAD 

MY COMRADE IN LIFE AND WORK 
THIS BOOK is DEDICATED 



PREFACE 

CLIMATE influences the surface of the earth, and this con- 
versely, in its varieties, determines the climate under other- 
wise equal conditions. This intimate mutual connection 
makes climatology and climatography appear as parts of geog- 
raphy, because they are essentially necessary to describe the sur- 
face of the earth and its changes. These ideas find their expression 
in the fact that generally in colleges and universities, climatology 
as a whole is treated in the geographical departments. Perhaps 
the dependent role of climatology may be attributable also to the 
fact that geographers have so greatly furthered this science. 

For the most part, climatographies have been written by ge- 
ographers. Therefore, geographical methods are kept in the fore- 
front, and specifically climatological methods are not so much used. 
It would be satisfying if this book offered a bridge connecting the 
two realms. 

To judge from the author's many years of experience in Europe, 
followed by a few years in the United States, the student in ge- 
ography has perhaps sufficient knowledge of climatology, but he 
does not know how to deal with the original data of observations. 
He is, I should say, familiar with the results but not with the ways 
of getting them. This holds, particularly, for the different methods 
of mathematical statistics which can be adapted to an individual 
representation of the climates; only thus can too-schematical 
descriptions be avoided. 

This book should make the student acquainted with a number 
of methods of mathematical statistics and theory of probability 
applicable at once to climatological problems. Approximations 
are used as much as possible in order to facilitate arithmetic. The 
student who is interested in the mathematical proofs and deriva- 
tions as well as in their philosophical background must be referred 
to the special technical literature. 

The general introduction presents climatology as a world 
science, and its international organization. The number of ob- 
servations in the meteorological register makes the necessity of 
statistical methods evident. Their special application to clima- 
tological series is discussed. An instruction, easy to understand, 



vu 



viii METHODS IN CLIMATOLOGY 

is given for computing periodical phenomena by means of harmonic 
analysis. 

In the foregoing, the overwhelming role of the frequency distri- 
bution has been emphasized. This idea led L. W. Pollak to intro- 
duce a new system of treating climatological observations by means 
of automatic statistical machines. This system covers the entire 
body of the observational data and may represent the future of 
rational climatological statistics. Pollak's method is not dis- 
cussed here, however, because only with the resources of great 
institutions is it practicable. 

The general representation of climatological elements, factors, 
etc., is followed by discussions of the special characteristics of the 
single elements. 

The radiation elements are not dealt with. As far as frequen- 
cies and different correlations with other elements are concerned, 
the methods are identical with those applied to other climatological 
elements, on the one hand. On the other, such a discussion would 
have had to include the physical and astronomical nature of the 
subject. This exceeds the range of this book. 

The first two parts of the preseat book are concerned with the 
variations of the elements in the course of time at one fixed place. 
The third part presents the comparison of the elements which are 
observed synchronously at different places, and arrives at their 
geographical distribution. 

Critical scrutiny of the observational data leads to the examina- 
tion of the homogeneity of climatological series and to the reduc- 
tion to an identical period. These problems remind me of Hann's 
quotation of Francis Bacon, the great English philosopher: Si quis 
hujusmodi rebus ut nimium exilibus et minutis vacare nolit, imperium 
in naturam nee obtinere nee regere poterit. 1 

These efforts are of fundamental importance not only for the 
climatologist but also for everybody who is interested in clima- 
tological series, as are the long-range forecaster and those dealing 
with periodicities and tree-growth analysis. 

Comparisons of data at different places reveal the concept of 
coherent and non-coherent climatic regions and of the climatic 
divide. These ideas show also clearly that the theory of correla- 
tion in the realm of climatology should play a significant role. An 
intermediate chapter deals with this subject. 

1 "A scholar who is unwilling to take pains over such investigations because they 
seem too insignificant and microscopic will be able neither to gain nor to maintain 
mastery over nature." 



PREFACE ix 

Climatological examples demonstrate the computations so that 
everyone with high-school training should be able to under- 
stand the mathematical procedure. Linear correlation, simplifica- 
tions of the calculation, and regression equations are discussed. 
Finally, the formulas of partial correlation of three variants are 
added. Rather much space is devoted to graphical methods of 
representation, especially in connection with mountainous regions. 
The great importance of anomalies and isanomals is shown and 
supported by examples. At the end of this chapter, air-mass 
climatology, continentality, etc., are mentioned. 

The fourth section gives suggestions for the arrangement of a 
more or less complete climatography. 

In the appendix, models for climatic tables are presented, a 
table for easy calculation of the probable error, an auxiliary table 
for computing the equivalent temperature, and a table giving the 
numbers of the consecutive days of the year. 

The content and order of treatment in this book reflect, for the 
greater part, the experience the author acquired in his lectures in 
climatology and in supervising the theses of his pupils at the 
University of Vienna, Austria. A number of the examples are 
taken from the author's papers. 

It is with the greatest pleasure that the author expresses his 
gratitude to Professor Charles F. Brooks of Harvard University 
for his repeated encouragement to summarize in book form the 
methods of climatology. Thanks are also due to Dr. Brooks for 
his reading of the manuscript and many suggestions as well as for 
constant help in facilitating the routine work connected with 
this book. 

The author wishes to express his appreciation for financial 
assistance from gifts of the Associates of Physical Science and 
Anonymous Fund 15 of Harvard University. 

He is deeply indebted to Professor L. W. Pollak, Dublin, Ire- 
land, who himself has contributed so much to the problem of 
statistical methods in meteorology and climatology, for his reading 
the manuscript and his valuable comments and suggestions. 

His warmest thanks are extended to Professor W. M. Fuchs, 
New York, Dr. J. T. Morley, Dublin, Ireland, and Dr. C. Chap- 
man, Boston, who did their best to smooth his English. Grate- 
fully he acknowledges the assistance of Miss Helen Gilman for 
redrawing the figures. Mr. R. W. Burhoe, Acting Librarian of 



x METHODS IN CLIMATOLOGY 

Blue Hill Meteorological Observatory, provided the necessary 
literature, always very helpfully and promptly. 

Owing to the present war, it has been impossible to obtain 
permission from the respective scientists and publishers to repro- 
duce here a number of graphs. The author offers his sincere 
apologies for this omission and assures all concerned that, in normal 
times, he would have waited until he had received permission, 
before publishing. In this book, he has at least been careful to 
see that in the legend every figure taken from another work has 
been duly attributed to the scholar who originated it. 

V. C. 

Cambridge , Mass. 
October l t 1944 



CONTENTS 
INTRODUCTION 1 

PART I 

GENERAL METHODS 
I. CLIMATOLOGICAL ELEMENTS. COMPARABLENESS OF CLI- 

MATOLOGICAL SERIES 3 

1. Climatological collaboration and organization. Sta- 

tions of different orders. The "rseau mondial" 5 

2. The meteorological register 9 

3. Some remarks about methods of climatological ob- 

servations 12 

a. Discontinuities in climatological series 12 

b. Observing air temperature. Errors caused by 

the influence of radiation and by layering of 
air close to the ground 13 

c. Measuring precipitation 15 

II. GENERAL STATISTICAL CHARACTERISTICS OF CLIMATOLOG- 
ICAL ELEMENTS 17 

1. The system of characteristics 17 

2. Primitive characteristics 17 

3. The variate. The absolute extremes. The range of 

variation 17 

4. Class intervals. Frequency distribution. Absolute 

and relative frequency 19 

5. Histogram and frequency curve. The mode. The 

modal class 21 

6. Elementary characteristics. Median, quartiles, dec- 

iles 24 

III. SPECIAL STATISTICAL CHARACTERISTICS OF CLIMATOLOGICAL 

SERIES 30 

1. The arithmetical mean 30 

2. Deviations. Average variability. Standard devia- 

tion. Normal distribution 31 

3. A practical application of the standard deviation in 

climatology 35 

4. Variance. Coefficient of variation 38 

5. Summary of statistical concepts 39 

6. Relation between ju, cr, and f. Cornu's theorem. 

Unilaterally and bilaterally limited variates 39 

xi 



xii METHODS IN CLIMATOLOGY 

7. Higher characteristics (skewness) 44 

8. Different kinds of variability used in climatology. . . 46 

9. Absolute and relative variability and other measures 

of variations 51 

10. Some applications of the method of random samples 53 

a. Precipitation 53 

b. Cloudiness and duration of sunshine 55 

IV. SOME PROBLEMS OF CURVE FITTING AND SMOOTHING OF 

NUMERICAL SERIES 58 

1. The straight line 58 

2. Decay-curves 62 

3. The equation y = B -e Ax 63 

4. The quadratic equation 64 

a. Level of the turning point of the curve 66 

b. The standard distribution 67 

5. Smoothing of numerical series 68 

V. HARMONIC ANALYSIS 70 

1. The analysis 70 

2. Evaluation of the equation 76 

3. Relative amplitudes 78 

4. Times of the extremes 78 

PART II 

REPRESENTATION OF CHARACTERISTIC FEATURES 
OF DIFFERENT ELEMENTS 

VI. TEMPERATURE 81 

1. Hours of observation. Reduction to the "true 

mean" 81 

2. Dates at which the average temperature crosses cer- 

tain thresholds 92 

3. Duration of temperatures above and below certain 

thresholds 94 

4. Other characteristics of temperature conditions. 

Spells of cold and hot days 95 

5. Cumulated temperatures 96 

6. Degree days 97 

7. Relative temperatures 98 

VII. ATMOSPHERIC PRESSURE AND PRESSURE OF WATER VAPOR 100 

VIII. WIND 101 

1. Resultant wind velocity calculated from records of 

direction and velocity 101 

2. Resultant wind direction calculated from frequencies 

of the directions alone 103 

3. The resultant run of the wind 104 

4. The steadiness of the wind 106 



CONTENTS xiii 

IX. SOME COMBINED ELEMENTS 107 

1. Relative humidity 107 

2. Equivalent temperature 108 

3. Drying power Ill 

4. Cooling power 112 

X. CLOUDINESS 115 

1. Clear and cloudy days 115 

2. Fog 117 

XI. PRECIPITATION 118 

1. Probability of a day with precipitation 118 

2. Probability that a day with precipitation is a day 

with snowfall 119 

3. Amounts of precipitation within certain periods. ... 119 

4. The annual course of the amounts of precipitation. . 120 

a. The reduction to months of equal length .... 120 

b. Amount of precipitation on an average day of 

the month 121 

c. Ecart pluviom&rique relatif and the relative 

pluviometric coefficient 121 

5. Absolute and relative variability. Quotient of varia- 

tion. Method of random samples applied to rec- 
ords of rainfall 122 

6. Rain intensity 122 

7. Wet and dry spells 123 

8. Characterization of the hydrometeoric type of a 

period 1 24 

9. Snow 124 

10. Index of aridity 127 

11. Rain-histogram 127 



PART III 
METHODS OF SPATIAL COMPARISON 

XII. COMPARISON OF OBSERVATIONAL SERIES OF DIFFERENT 
PLACES. GEOGRAPHICAL DISTRIBUTION OF CLI- 
MATOLOGICAL ELEMENTS 129 

1. Uniform tendency of the variations of average 

weather over large regions. Consequences drawn 
from this tendency for comparing series of ele- 
ments observed simultaneously at different places 129 

2. Criteria of relative homogeneity 134 

a. Helmert's criterion 134 

b. Abbe's criterion 135 



xiv METHODS IN CLIMATOLOGY 

3. Reduction of climatological averages to a certain 

period 139 

a. Method of differences 139 

b. The reduction of precipitation series. Method 

of quotients 142 

c. Physical explanation of the method of quo- 

tients 143 

d. The reduction of other elements to a given 

period 144 

e. Limits of the method of reducing climatological 

series to a normal period 144 

f. Additional note. The length of a normal 

period 146 

g. Interpolation of missing observations 148 

h. Coherent and incoherent climatic regions and 

their statistical evidence 148 

C. INTERMEDIATE CHAPTER. CORRELATION 150 

1. Linear correlation 150 

2. Example of calculating a correlation 153 

3. Simplifications in computing correlations 155 

4. Regression equation 157 

5. Partial correlation \ 161 

XIII. GRAPHIC COMPARISON OF CLIMATOLOGICAL ELEMENTS. 

ISOGRAM. MAP OF ISOLINES 166 

XIV. METHODS OF ANOMALIES 170 

1. Temperature distributions in mountainous regions. 170 

2. Examination of lapse rate 170 

3. Mapping temperature conditions of a mountainous 

country (Reduction to a given level) 171 

4. Actual temperatures 172 

5. Standard curves. Anomalies. Isanomals 173 

6. Applications of the method of anomalies and 

isanomals 1 74 

a. Vegetative period and altitude 1 74 

b. Temperature and geographical latitude 177 

c. Pleions, meions 178 

d. Snow-cover and altitude 180 

e. Duration of sunshine vs. elevation 180 

f. Anomalies of precipitation 180 

7. Precipitation profiles 183 

8. Variability of precipitation and amounts of precipi- 

tation 185 

XV. WIND 186 

1. Wind roses 186 

a. Wind roses for different elements 187 



CONTENTS xv 

2. Maps of streamlines 188 

a. Streamlines based only upon frequencies of 

direction 192 

XVI. AlR MASS CLIMATOLOGY 193 

XVII. NUMERICAL CHARACTERIZATION OF DIFFERENT CLIMATIC 

FEATURES 195 

1. Continentality 195 

2. Limits between timber-forest, steppe, and desert. 

Effectiveness of precipitation 196 

PART IV 
THE CLIMATOGRAPHY 

XVIII. ARRANGEMENT OF A CLIMATOGRAPHY 199 

1. The introduction 199 

2. Static climatology 200 

3. Monographs 201 

4. Records of self-registering instruments 202 

5. Change of temperature and precipitation with height 202 

6. Graphical representation 202 

7. Dynamical climatology 206 

8. The bioclimatological part 212 

9. Description of climatic phenomena 212 

XIX. CONCLUSION 215 

APPENDIX I. Forms of individual climatic tables 216 

APPENDIX II. Vl/n 218 

APPENDIX III. Auxiliary table, calculating the equiva- 
lent temperature from air temperature 

and relative humidity 219 

APPENDIX IV. The days of an ordinary year, numbered 

consecutively, beginning January first. 220 
INDEX 221 



TABLES 

1. Daily temperature minima at Mt. Washington, N. H., for four 

months of January 18 

2. Frequency table and class intervals 19 

3. Frequency distributions characterizing different climates 20 

4. Example of an array and grouped data 24 

5. Frequency distribution for a larger sample 25 

6. Dates of the last killing frost. Deviations 32 

7. Classification of temperature minima based on the probable 

error 37 

8. Statistics of rainfall in March at Helwan, Egypt 42 

9. Examples of average variability and inter-sequential varia- 

bility 48 

10. Different statistics regarding interdiurnal variability of tem- 

perature 49 

11. Method of random samples applied to rainfall 55 

12. Examples illustrating the relation between duration of sunshine 

and cloudiness 56 

13. Annual amplitude of air pressure and altitude 58 

14. Example of observations represented by an exponential curve. 63 

15. Example of observations represented by a quadratic equation. 65 

16. Clipping from a table of normal distribution of temperature in 

a mountainous country 67 

17. Instruction for calculating the harmonic constants for 12 equi- 

distant values 73 

18. The same for 24 equidistant values 73 

19. Example of the harmonic analysis of an annual course 76 

20. Evaluation of the resulting equation 77 

21. Example of a daily course of temperature 82 

22. Hours of observations at climatological stations in the United 

States 84 

23. Examples of the annual course of the correction to give the 

"true mean" for different hours of observation 85 

24. The correction term of the arithmetical mean of the average 

extreme temperatures to give the "true mean" 89 

25. The annual course of temperature at Bismarck, N. D 93 

26. Average cumulated daily minimum temperatures for Paris. ... 96 

27. Examples of degree days 97 

28. Examples of the calculation of the wind elements 101 

29. Equivalents of the numbers of the Beaufort scale 105 



XVll 



xviii METHODS IN CLIMATOLOGY 

30A. Average monthly equivalent temperatures at Boston, Mass., 

computed by the approximating formula 110 

30B. Comparison between the approximating and the correct formula 1 10 

31. Frequency of clear and cloudy days and cloudiness 116 

32. Probability that a day with precipitation is a day with snowfall 119 

33. Representation of the annual course of precipitation 120 

34. Form for representing snow conditions 125 

35. The quasi-constancy of differences of temperature at two differ- 

ent places 131 

36A. The quasi-constancy of the ratios between rainfall at two differ- 
ent places 132 

36B. Fluctuations of rainfall amounts as percentage of the average 
values as illustrative of the quasi-constancy of ratios (BjW) 
(after L. W. Pollak) 133 

37. Example of the application of Abbe's criterion to differences of 

temperature 137 

38. The same for ratios of rainfall 137 

39. Temperature conditions for different periods at St. Paul, Minn. 139 

40. Rainfall on the Maiden Island during different periods 139 

41. Example of calculating a correlation 154 

42. Calculating a correlation by means of the original numbers 

diminished for a constant value 156 

43. Another example of calculating a correlation 159 

44. Standard distribution of the duration of the vegetative period 

in Switzerland 175 

45. Examples of anomalies 175 

46. Standard distribution of temperature with geographical latitude 177 

47. Example of wind roses 188 

48. Degree of continentality derived from frequencies of continental 

and maritime air masses 193 



ILLUSTRATIONS 

FIGURE 

1 . Structure of a climatological network 6 

2. The international climatological register 10 

3. The register of the cooperative stations (U.S.A.) 11 

4. Frequency histogram and curve 22 

5. Normal distribution and standard deviation 34 

6. Frequency distribution of daily minimum temperatures. ... 38 

7. Frequency curve of a unilaterally limited variate 43 

8. Frequency distribution of a bilaterally limited variate 43 

9A. A right-skewed frequency distribution 45 

9s. A left-skewed frequency distribution 45 

10. Relative variability and rainfall 52 

11. Cloudiness and duration of sunshine 55 

12. Annual range of air pressure and altitude 59 

13. Observations represented by the equation y = Ax~ B 62 

14. Observations represented by the equation x= A + By + Cy 2 64 

15. Superposition of two waves 71 

16. Determining the true phase angle 75 

17. Evaluation of a Fourier-series of two terms 77 

18. Daily course of temperature 82 

19. Isolines of the correction of the averaged daily extremes of 

temperature to give the " true mean M for the United States 

in January 90 

20. The same for July 90 

21. Interpolating the dates at which the annual temperature 

curve crosses certain thresholds 92 

22. Relative temperatures 98 

23. Evaluating the true azimuth of the resultant wind 103 

24. Frequency of wind directions and their average velocities. . . 104 

25. Climogram of Boston, Mass 107 

26. Variations of January temperatures at Moscow 130 

27. Extreme range of temperature and length of the period of 

observation 147 

28. Dot-diagram and regression lines 161 

29. Isopleths 166 

30. Isanomals of the duration of the vegetative period in Switzer- 

land 176 

31. Isanomals of January temperatures 179 

32. Isanomals of sunshine duration for the eastern Alps in winter 181 

33. Profile of precipitation across the Bernese Alps 184 

34. Profile of snowfall across New England 185 



XIX 



xx METHODS IN CLIMATOLOGY 

FIGURE 

35. "Wind-star" 187 

36. Streamlines over the Balkans 189 

3?A. Streamlines of New England in January 190 

37e. Streamlines of New England in July 191 

38. Isolines of the dates when the annual temperature curve 

exceeds 70F in the Mediterranean region 204 

39. Isolines of the dates when the annual temperature curve 

drops below 70F in the Mediterranean region 204 

40. Isolines of the duration of 70F in the Mediterranean region 205 

41. Isolines of the duration of the dry period in the Mediter- 

ranean region 205 

42-45. Distribution of the basic elements characterizing normal, 

cold, and warm seasons in the United States 207-210 

46. Eroded slopes in the mountains of Albania facing p. 212 



METHODS IN CLIMATOLOGY 

INTRODUCTION 

CLIMATOGRAPHY describes the climate, climatology explains it. 
Climatography prepares the raw material supplied by ob- 
servation; it is the foundation of climatology. This seems 
a vicious circle, because it is hardly possible to make a climatog- 
raphy without a knowledge of climatology. This is the present 
state in which the two branches of science supplement each 
other. In its early beginning, climatography was an occasional 
description in words and pictures by travelers, chroniclers, writers, 
poets, and painters. 

The invention of the thermometer and its regular use for ob- 
serving air temperature was the first step toward a quantitative 
climatography. Galileo Galilei invented the thermometer in 1597. 
It became an accurate physical instrument in 1780 when de Luc 
introduced mercury as a thermometrical substance. Rain gauges 
are supposed to have been used in India in the fourth century B.C. 
The oldest regular quantitative measurements of rain were made 
in Palestine in the first century A.D. With these observations at 
hand, it has been possible to compare the hydrometeoric climate of 
ancient Palestine with the present one in an exact way. The 
invention and development of other instruments has furthered 
regular quantitative climatological observations. Elements which 
cannot be observed by instruments are estimated by means of 
arbitrary scales. 

A single observation, picked from a series, has usually no clima- 
tological meaning. Only the chronological arrangement and, gen- 
erally, the combination of observations, are sufficient basis for 
computing average variations and average states. A single ob- 
servation can be compared to a single exposure of a movie film, 
which has no real meaning until it is combined in its proper sequence 
with the rest. 

The nature of our subject determines the necessary analytical 
and synthetical methods. First of all, the meaning of climate may 
be explained : 

Climate is the average state of the atmosphere above a certain place 
or region of the earth's surface, related to a certain epoch and consider- 



is subject. 1 

Observations are made at isolated points. Only by comparing 
these data can the climate of the whole region be interpolated. 
It is therefore the principal and fundamental aim of climatological 
methods to make the climatological series comparable. The more fully 
this goal is approached the more reliable are the indications of a 
climatography. 

Long-range forecasts or investigations in hidden periodicities 
depend even more on reliability and exactness, and therefore upon 
the comparableness of climatic series. Only with comparable 
climatic series can true averages, true variations, true probabilities 
be determined and used for making further estimates and inferences. 

1 V. Conrad, " Die klimatischen Elemente und ihre Abhangigkeit von terrestrischen 
EinflQssen," Handbuch der Klimatologie, ed. by W. Koppen and R. Geiger, vol. IB 
(Berlin, 1936). 



PART I 
GENERAL METHODS 

CHAPTER I 

CLIMATOLOGICAL ELEMENTS. COMPARABLENESS OF 
CLIMATOLOGICAL SERIES 

CLIMATOLOGICAL ELEMENTS result from the analysis of the 
state of the atmosphere. The combination of all elements 
occurring at a given moment makes the weather; the average 
weather means the average state of the atmosphere, that is, the 
climate. 

It is not possible to enumerate all the elements, because their 
number can be increased arbitrarily, but the following may be 
listed : 

1) Radiation of sun and sky 

2) Temperature of the air and of the surface of the earth 

3) Wind direction and velocity 

4) Humidity and evaporation 

5) Cloudiness and sunshine 

6) Precipitation 

7) Snow cover 

8) Air pressure, because of its intimate relation to the 

instantaneous and average state of weather and 
atmosphere. 

The difficulty is that these items do not represent single ele- 
ments, but groups of elements. The radiation of sun and sky, 
as an example, is divided into two main groups: direct radiation 
from the sun, and radiation from the sky. Each group can be 
subdivided into any number of elements. There is the total 
radiation, including the energy of the entire spectrum, which can 
be divided into the elements characterized by the energy of parts 
of the spectrum, for example the infrared, the visible, the ultra- 
violet portions. Other radiation elements deal with the trans- 
mission coefficients of certain wave-lengths, with the turbidity 
factor, the blueness of the sky, photometrical and photochemical 



4 METHODS IN CLIMATOLOGY 

brightness, atmospheric back radiation, range of the ultraviolet 
spectrum, etc. 1 

The group of radiation represents only one example. The same 
conditions exist with all the other groups. This nearly infinite 
multifariousness is no misfortune. On the contrary, a clever 
climatographer, while utilizing the usual elements, should define 
new elements which give the best, the shortest, and the most accu- 
rate contribution to the description of the climate in consideration. 

For the present purpose, it is perhaps of greater importance to 
classify the elements from another point of view : 

1) Primitive elements, directly observed or estimated 

2) Combined elements 

3) Derived elements. 

Examples of group 1, primitive elements, are temperature, precipi- 
tation, wind velocity, wind direction, cloudiness, etc. Examples 
of group 2, combined elements, are turbidity factor, continentality, 
equivalent temperature, potential waterpower, etc. 

There are also elements which may be taken both as primitive 
and combined elements; examples are relative humidity and cooling 
power. On the one hand, each can bfc read directly from its ap- 
propriate instrument; on the other hand, each can be calculated 
by means of a combination of two or three other elements. 

The third group is obviously the most extensive and offers much 
scope to the imagination of the climatographer and climatologist. 
Only a few examples may illustrate the importance of this group. 
There are the variabilities of the different elements, the length of 
certain spells, as of high and low temperatures, of dry or wet 
weather, etc. ; the duration of characteristic average states of the 
atmosphere for example, the duration in days of the period during 
which the average temperature is below the freezing point (frost 
period) or the duration of the vegetation period, that is the time 
during which the average temperature is ^43F; the lapse rate of 
temperature; the manifold use of frequencies and probabilities, 
anomalies, etc. 

From the classification of the elements as primitive, combined, 
and derived, their infinite multiplicity is obvious; 2 hence the need 

1 For the multiplicity of radiation elements, see the monograph of F. W. P. G6tz, 
Das Strahlungsklima von Arosa (Berlin, 1926). From the same point of view, W. 
M6rikofer, " Meteorologische Strahiungs Messmethoden," in Emil Abderhalden, Hand- 
buck der biologischen Arbeitsmethoden (Berlin, 1920 ) Abt. II, pt. 3, is of interest, 

1 In the foregoing series of climatic elements, the one or the other is enumerated 
which may be unknown to the reader. Some of them are explained in the following 



CLIMATOLOGICAL ELEMENTS $ 

of some methodical hints on handling these numerous expedients 
and selecting the best in order to get the most concise and exact 
description of the climate. 

Observations do not always reproduce natural conditions in the 
right way. There are many sources of error. An appropriate 
critique of the data should eliminate the systematic as well as 
random errors. These corrections concern a series of observations 
made at one place for a period of time, and a comparison of 
observations made simultaneously at different places. 

The preparation of original observations for purposes of com- 
parison, whether as single values or as a series, is one of the main 
tasks of climatological methods. 

I. 1. CLIMATOLOGICAL COLLABORATION AND ORGANIZATION. 

STATIONS OF DIFFERENT ORDERS. THE 

"RESEAU MONDIAL." 

Climatology is a world-wide science. That is clear from the 
standpoint of comparability. The observations of a station A 
have to be compared with those of station B south of A ; the ob- 
servations of C west of A should be compared either with A or 
with B, and so forth, for all directions of the compass. (See Fig. 1.) 
Thus we may understand why Sir Napier Shaw writes in his 
Manual of Meteorology (I, 160) :"No country, not even the largest, 
is self-sufficient in the material for study, because that material 
must be co-extensive with the whole world." 

The first series of observations were made by private men and 
institutions interested in weather and nature. In 1781, the meteor- 
ological organization the " Mannheimer Akademie" was founded; 
and this was the first step toward a world-wide climatography. In 
their publications, Ephemerides Societatis Meteor ologicae Palatinae, 
the results of observations for two American places were also 
published. 

Nowadays, in civilized countries, the State has organized central 
institutes which administer and extend the network of climato- 
logical (meteorological) stations. The expression " network " is 
taken from the "network of stations " upon which the geodetic 
survey is based. Figure 1 indicates more fully the meaning of the 

chapters in so far as they are of methodological interest. Other explanations can be 
found in V. Conrad, Fundamentals of Physical Climatology (Harvard University Press, 
1942) and in V. Conrad, "Die klimatologischen Elemente und ihre Abhangigkeit 
von terrestrischen EinfKissen," Kdppen-Geiger, Handbuch der Klimatologic, vol. IB 
(Berlin, 1936). 



6 METHODS IN CLIMATOLOGY 

term. The observations of each station have to be comparable 
with those of the others. The aim of the study of the reciprocal 
relations between the climatic elements at different places is, on 
the one hand, to determine the circulation of the atmosphere 
(physics of the actual state of the atmosphere), and, on the other, 
to get full information about average and extreme values of clima- 
tological elements at least on the earth's surface (physics of the 
average state of the atmosphere). 




FIG. 1. Structure of a climatological network 

For these large-scale purposes, a network, internationally ad- 
ministered, has to cover the entire globe. Regional meteorological 
efforts found their international organization at a relatively early 
stage. An international conference of maritime meteorology was 
held at Brussels in 1853, ninety years ago, and the first international 
meteorological congress at Vienna (Austria), in 1873, was attended 
by official delegates from a large number of countries. On this 
occasion an " International Meteorological Committee " was elected 
in order to assure international meteorological and climatological 
collaboration without interruption during the interval between the 
congresses. 

This international organization has the following duties : 

1) To make regulations and offer advice concerning com- 
parable observations. 



CLIMATOLOGICAL ELEMENTS 7 

2) To establish definitions of phenomena like rime, snow- 

cover, day with thunderstorm, day with precipita- 
tion, etc. 

3) To make regulations regarding the organization of a 

network for a country. Such a network should consist 
of stations of different orders : 

(a) A Central Office, or Central Institute, is the chief office en- 
trusted by the Government with the management, collection and 
publication of the meteorological observations of the country 
[e.g. the U. S. Weather Bureau]. 

(6) A Central Station [Section Center] is a subordinate center 
for the management and collection of observations from a certain 
province [state; e.g. the U. S. Weather Bureau, Boston, Mass.]. 

(c) A Station of the First Order is an observatory in which, with- 
out the collection of observations from other stations, meteor- 
ological observations are conducted on a great scale, i.e. either by 
hourly readings or by the use of self-recording instruments. 

(d) Stations of the Second Order are the stations where com- 
plete and regular observations on the usual meteorological ele- 
ments, viz., pressure, temperature and humidity of the air, wind, 
cloud, rain and hydrometeors, etc., are conducted. 

(e) Stations of the Third Order, finally, are the observing sta- 
tions, where only a greater or less portion of these elements are 
observed. 3 

A knowledge of the international organization, of its regulations 
and its definitions, is of great importance to anyone who has to 
use the meteorological records of other countries. The most es- 
sential resolutions and recommendations of the International 
Meteorological Committee during the years from 1872 to 1905 are 
summarized in the International Meteorological Codex. 4 In spite 
of the age of this report (published in English by the British 
Meteorological Office in 1909), the recommendations are valid and 
valuable today with the exception of the sections on aerological 
observations and on weather maps and forecasting. A Spanish 
version was published by the Observatorio Central de Manila in 
1913, and includes the resolutions adopted between 1872 and 1910. 5 

Another merit of the International Meteorological Organiza- 
tion is the publication of the reseau mondial, a global network of 
meteorological stations. This title represents the climatological 
and meteorological ideal: a network (as stated above), which 

1 Sir Napier Shaw, Manual of Meteorology, I (1926), 163. 
4 Prepared by G. Hellmann and H. H. Hildebrandsson. 

*Sir Napier Shaw, Manual of Meteorology, I, 166. For further information, see: 
Official Reports of the International Conferences. 



8 METHODS IN CLIMATOLOGY 

covers the entire surface of the earth, administered in a uniform 
manner and devoted to all of mankind. The publication in ques- 
tion is only the beginning of an ideal as yet unaccomplished. It 
was a great scientific advance when the British Meteorological 
Office initiated it in 1911. Monthly averages of pressure, tem- 
perature, and precipitation at about five hundred places are pub- 
lished for each year. The stations are grouped in 10-zones of 
latitude. It is noteworthy that meteorological data can be sup- 
plied for all the zones from 80N to about 61S. South of this 
latitude no permanent meteorological station is being operated. 

The data and elements given for each station in the rSseau 
mondial for the months and the year are : 

1) Latitude 

2) Longitude 

3) Height (meters) 

4) Pressure (millibars): 

(a) Mean at the level of station 

(b) Mean at mean sea level (M.S.L.) 

(c) Differences from normal 

5) Temperature in the absolute scale, starting from 273C 

below the freezing point of water: 

(a) Mean maximum 

(b) Absolute maximum with date of occurrence 

(c) Mean minimum 

(d) Absolute minimum with date of occurrence 

(e) Arithmetical mean of (a) and (c) 

(f) True average temperature 

(g) Difference from normal 

6) Precipitation: 

(a) Monthly amount in millimeters 

(b) Difference from normal 

Another document of the international development of clima- 
tology and meteorology is the World Weather Records, assembled 
and arranged by H. Helm Clayton, in consequence of a resolution 
of the International Meteorological Conference adopted at the 
meeting of Utrecht in 1923. This publication 6 gives for each year 
of record the monthly and annual values of air pressure, tempera- 
ture, and precipitation, and averages of all, for about 390 places. 

e Smithsonian Miscellaneous Collections, vol. 79 and vol. 90, Washington, D. C. 
1927 and 1934. 



CLIMATOLOGICAL ELEMENTS 9 

The great advantage is that the publication offers long and, as far 
as possible, homogeneous series, which overlap one another for 
the period of observation. Thus the work is indispensable if 
variations of the three elements have to be compared. 

I. 2. THE METEOROLOGICAL REGISTER 

The meteorological register contains the records of the original, 
regular observations made at the climatological station. All cli- 
matological studies are based ultimately upon the registers of the 
stations. 

The utmost international uniformity of the registers would be 
advantageous for all climatological investigations which cover a 
region larger than one national network. This was the reason 
why the International Meteorological Organization (Utrecht, Hol- 
land, in 1874) suggested a special form for the register of stations 
of the second order. Figure 2 shows this international form with 
a few supplements. As far as the units of pressure and altitude, 
the temperature scale, etc., are concerned, uniformity could not 
be achieved. 

This form, which assumes that the elements mentioned are 
observed three times a day, is used by most of the central institutes 
of Europe, of the U.S.S.R., etc. In many networks, the register 
is more or less modified, according to the various conditions and 
requirements. It is to be hoped that these modifications will be 
restricted more and more, and will give way to complete uniformity. 

As far as the organization of the United States Weather Bureau 
is concerned, J. B. Kincer may be quoted: 7 

The Weather Bureau has maintained some 200 observing stations 
known as stations of the first order, manned by professional meteor- 
ologists, at which complete meteorological observations are made, in- 
cluding automatic, continuous instrumental records of many weather 
elements. Since these stations are not spaced closely enough to provide 
data adequate for climatic purposes, a large number of cooperative 
stations . . . are maintained to obtain the necessary basic data. [The 
number of these cooperative stations is about 5,000.] On the other hand, 
the U. S. Weather Bureau has branches in the States which administer 
the respective climatic section and are equipped with many instruments, 
especially automatic-recording instruments. 

7 "Climate and Weather Data for the United States/' in Climate and Man (Year- 
book of Agriculture, United States Department of Agriculture, Washington, D. C. f 
1941), p. 689. 



10 



METHODS IN CLIMATOLOGY 



II H 

4? 43 43 



c 





a 

4) 

S 



o 

S 



l 



a 



43 
43 



43 
43 
43 



43 
43 



Remarks about 
hours and 
duration of 
precipitation, 
thunderstorms, 


storms, optical 
phenomena, etc. 




d^ 






^* 






43 &g 

4- ^ 






Q 






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cl crJ <n 






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6 



CLIMATOLOGICAL ELEMENTS 



11 



Because the network of the cooperative stations of the U. S. 
Weather Bureau is one of the greatest in the world, it may be con- 
venient to give here the register of these stations as an example of 
a somewhat modified international form (Fig. 3). 



U. S. DEPARTMENT OF COMMERCE, WE ATHER BUREAU 
COOPERATIVE OBSERVERS' METEOROLOGICAL RECORO 



snnont <fr*tfri//f . . , C(. 

t ATirunr44 f 23 A/ LM9/ruOf74'27W fLCUirtO*. 
ttONTM WA*/- , 1934. HfltlDtAN Of TIME. 



t-Jte 



27. 



'MT 



21 






MONTHLY SUMMARY 



MICH WINOt, JffM. ,. 

GlAZt. tmttiiM IUINI.. 



KltUNOfKOftT,. 

TMONOCMSTOMMS, 



HtfL 



MMC " kdt f M i /J//T..,, 



OCPTH Of rK02CN OMOUNO AT CNO Of MO _W1 



^3*5*7 



29 



4i 



1P- 



SI 



JLiL 



J^ 



(MOt TO M riCtCO M OtMHVl*) 

TEMPERATURE 

MfAN MAIMUM 
MCAN MINIMUM ^ 

MEAN MONTMtt. DCWkHTW 

MAXIMUM 
MINIM 



16 



TOTAt.- 



9 



PRECIPITATION 

INS OC*AaTU*f,_ 

QNCATCST AMOUNT 



f* 



47 



35 



1.5 



TOTAL SNOfTfALt. 

MCATCST IN ANY 00SCftV*TlONAt OAT. 



8 



M- 



tl+r **/* A fO* /M 



= *. 



NUMOEROPOAYS 
MAX TCM^ O* **ft^> 
MAX TCM^ Jl 0* "*. 
MIN TtM^ S ON ItLO* 

MIN TCMP o* o* ICLOI 

WITH 01 INCH 0* MOC PKCCl^. 

l INCH on MOKE, . IOO tNCH Oft MOftC, 

CLt A. MUTLT CLOUOT,___ttOUOt. 




FIG. 3. Form of the register of the cooperative stations of the 
U. S. Weather Bureau 

The greatest difference between the two forms lies in the fact 
that the international form is arranged for three observations a 
day, while the U. S. cooperative observer makes one observation 
a day. Two columns are reserved for the extremes of temperature. 
A third column of this section contains numbers which are not of 
interest here. A great advantage, not provided in the interna- 
tional form, is the columns which indicate the time of the be- 
ginning and end of precipitation. Fundamentally important differ- 
ences are noted in the case of cloudiness and wind : here the prin- 



12 METHODS IN CLIMATOLOGY 

ciple of making observations of an instantaneous state is not 
maintained, estimates of an average state during the day preceding 
the observation being given instead. 

The remarking of differences does not imply, of course, a criti- 
cism; a scientific organizer is free to arrange the methods of ob- 
servation so that he gets the greatest number of features character- 
istic of weather and climate. The student who builds up his 
climatography on the registers should be conscious of these facts, 
even if he is not compelled to go back to these fundamental sources. 

It might happen, for instance, that the region whose climate 
has to be described is crossed by the borderline between two clima- 
tological networks. In the first of these, the average direction of 
the wind during the day is estimated; in the second, momentary 
wind directions are observed three times a day. It would be a 
source of errors if the different methods of observation escaped 
notice. 

Usually, a space in the form is reserved for some simple sta- 
tistical values which can be easily derived from the original 
observations. 

I. 3. SOME REMARKS ^BOUT METHODS OF 
CLIMATOLOGICAL OBSERVATIONS 

The following discussion is not intended as an instruction on 
the arrangement of climatological observations. It serves only to 
emphasize difficulties of comparison which may arise with different 
elements. 

I. 3. a. Discontinuities in Climatological Series 

When the variations with time of an element at a place have 
to be investigated, its values at different times are compared. 
Changing the method of observation can cause a break, a discon- 
tinuity of the series. If a thermometer of the ordinary type is 
read over a period of ten years and is then replaced by a sling- 
thermometer, the temperatures during daytime become lower and 
night temperatures become slightly higher. Thus a discontinuity 
is caused, so that the second series cannot be compared with the 
first. The paradox is that while the sling-thermometer means a 
real improvement, this improvement destroys the continuity of the 
series. If after the introduction of the new procedure parallel 
observations are made in the old and in the new manner over a 
sufficiently long period, the values of the old series can be reduced 



CLIMATOLOGICAL ELEMENTS 13 

to those of the new one. Suitable methods for such reduction are 
given in the following chapters. 

An example of a nearly ideal "secular" station is represented 
by the Blue Hill Meteorological Observatory, south of Boston, 
Mass., founded by Abbott L. Rotch in 1885. Here observations 
have been made with suitable devices according to well-considered 
methods from the start, fifty-eight years ago. While numerous 
more modern methods were being introduced, the old methods of 
observing the most fundamental elements were continued simul- 
taneously with the new ones. No change has taken place in the 
surroundings. 

I. 3. b. Observing Air Temperature. Errors Caused by the 

Influence of Radiation and by Layering of Air 

Close to the Ground 

Observing the air temperature in order to make comparisons 
involves a considerable number of difficulties. Only one example 
need be mentioned here, which may contribute to a better under- 
standing of the later sections of this book. A well-made ther- 
mometer shows the "true temperature of the air M if its bulb is in 
full heat-conductivity balance with the passing air. This ob- 
viously simple demand can hardly be fulfilled in free air because 
of the radiation from sun and sky, from the ground and from sur- 
rounding objects. The invention of the "aspiration-thermometer " 
eliminated the errors caused by radiation, for meteorological pur- 
poses. Because of some difficulties which need not be discussed 
here, climatological air temperature is defined by the indications of 
a well-constructed aspirated thermometer at the level of about 5 
feet above the ground. Thermometers sheltered in a suitably 
placed "Stevenson screen " show temperatures which, generally, 
equal the indications of an aspiration-thermometer, so that the 
difference can be practically neglected. Thus, but for the existence 
of another difficulty, the problem of observing the climatological 
temperature would have been solved. 

In the Pamirs (Central Asia, about lat. 38N), in summer, a 
difference of about 60 F 8 was observed between the temperature 
of the layer adjacent to the ground, and that of the 5| ft level 
above it. 9 Once in winter, in a depression in the eastern Alps, a 

The author gladly introduces a suggestion of Charles F. Brooks to discriminate 
in printing between F, C = an actual temperature and F, C = Fahrenheit, Centi- 
grade degrees representing a temperature difference. 

9 V. Conrad, "Die klimatologischen Elemente und ihre Abhangigkeit von terres- 
trischen Einfltissen," K6ppen-Geiger, Handbuch der Klimatologic, IB, p. 72. 



14 METHODS IN CLIMATOLOGY 

temperature of 20F was observed on the ground and, simul- 
taneously a temperature of +30F at the 180 ft level above the 
ground. 10 

The physical explanations of these conditions, interesting as 
they are, need not be set forth. Here, only the fact counts that 
thermal stratifications of enormous intensity exist within thin layers 
above the ground. Such abnormalities, though not of this in- 
tensity, are frequent everywhere. We know this from an investi- 
gation in an exceedingly equable climate, near London, England. 11 
From this study we also learn that the average vertical tem- 
perature gradient in Kew (England) is so great that the tempera- 
ture decreases 1 F per 3J inches, within a layer from 1 ft to 4 ft 
above the ground, in June. This is a gradient 625 times greater 
than the dry-adiabatic gradient. 

The stratification of the air according to its specific gravity is, 
however, a very common climatic phenomenon in the concavities of 
the surface, when the outgoing radiation exceeds the incoming. 
The stratification is observed for a concave surface of any dimen- 
sion. It appears in the broad and narrow valleys of the high 
mountains of the earth, and is experienced in every furrow of the 
field. Thus the layering of the f air is of interest, not only for 
general considerations in observing air temperatures, but also as 
one of the most important features from which "Microclimatol- 
ogy," and especially "Orographic Microclimatology," makes its 
start. 12 

And now a problem arises concerning the temperature-observa- 
tion : namely, to which layer should the observation of temperature 
be related, if the temperature varies in the lowest layers for every 
3^ inches in the vertical direction as much as in the horizontal for 
about one degree of latitude. The observations with the aspiration- 
thermometer, especially, become very inexact and ill-defined. 
Temperatures vary several degrees, according to the direction from 
which the air is sucked up. 

There is only one expedient. The place of observation at a 
climatological station should be a spot with good natural ventila- 
tion, open to all wind directions, so that thermal stratification is 
rare. Furthermore, the thermometer-bulb should not be too close 
to the ground. The International Meteorological Organization 

10 Ibid., p. 183. 

11 A. C. Best, Transfer of Heat and Momentum in the Lowest Layer of the Atmosphere 
(Geophysical Memoirs of the Meteorological Office, London, No. 65, 1925). 

12 R. Geiger, Das Klima der bodennahen Luftschicht (Braunschweig, 1927). See 
especially the second part: " Orographische Mikroklimatologie," and Figure 30, p. 98. 



CLIMATOLOGICAL ELEMENTS 15 

advises fixing the thermometers at a height between 5 and 6| feet. 
The thermometers in the screen of cooperative observers of the 
U. S. Weather Bureau are at about the height mentioned, and thus 
exclude the bottom layers with intense, vertical temperature 
gradients positive or negative. 

We learn from this example that the exposure of the thermo- 
meter shelter must be chosen so as to avoid, if possible, local influ- 
ences of radiation and thermal stratification. Generally, only the 
former influence is mentioned, but the latter is also a great dis- 
advantage. Both factors seriously impair the comparableness of 
the data. The thermometer shelter should be located so that the 
recorded temperatures are representative of the largest possible 
surrounding area, a condition which also makes for the most com- 
parable results. It is obvious how badly the continuity of a series 
of temperatures may be disturbed by shifting of the shelter. 

I. 3. c. Measuring Precipitation 

Measuring precipitation offers a difficult problem. None of the 
various methods gives entirely satisfactory results. The amount 
of water or snow collected in the gauge depends, to a certain degree, 
upon the wind velocity and the air resistance of the particles of 
precipitation. The stronger the wind and the greater the air 
resistance to the particles, the smaller is the amount caught in the 
gauge compared with that collected under otherwise similar condi- 
tions during absolute calm. Shielded gauges, invented by F. E. 
Nipher, give incomparably better results. 13 

In general, the normal climatological stations are not yet 
equipped with shielded gauges, so that the wind and the different 
types of precipitating particles still influence the measurements: 
hence, a degree of comparableness can be achieved only by using 
identical gauges installed in a uniform way. 

This is true for the United States, half a huge continent, \ here 
the exposure of rain-gauges follows sound uniform principles. If 
some reader is interested, however, in a study of precipitation in 
Europe, he is confronted with much more complex conditions. 

18 For further information, see: C. F. Brooks, "Windshield for Precipitation Gauges" 
(Transactions of the American Geophysical Union, 1938, p. 539); C. F. Brooks, "Further 
Experience with Shielded Precipitation Gauges on Blue Hill and Mount Washington" 
(ibid., 1940, pt. II, p. 482); H. S. Riesbol, "Report on an Exploratory Study of Rain 
Gauge Shields and Enclosures at Coshocton, Ohio" (ibid., p. 474); F. N. Denison, "A 
Report on the Difference Between the Precipitation Records as Taken on the Standard 
Canadian and U. S. Rain-Gauges" (Bulletin of the American Meteorological Society, 
February 1941, p. 65). 



16 METHODS IN CLIMATOLOGY 

From the following table, the areas of the receivers of types of 
rain-gauges used in some countries in Europe may be seen, and 
for comparison, the area of that used by the U. S. Weather Bureau : 

Austria 78 sq. inches 

England 31 sq. inches type I 

England 20 sq. inches type II 

France 62 sq. inches 

Germany 31 sq. inches 

U. S. Weather Bureau 50 sq. inches 

Perhaps the size of the receiver has no great influence upon the re- 
sults. In any case, these numbers show that there is no uniformity. 
As was said above, measurements of precipitation depend on 
wind velocity. Because this varies enormously with the height 
above the ground, it is clearly a scientific requirement that the 
rim of the receiver should be at the same height in all countries. 
In reality, the heights are: 

Austria 59 inches 

England 12 inches 

France 59 inches 

Germany 39 inches 

U. S. Weather Bureau 34 inches 

This is a violation of the scientific requisite of uniformity. The 
climatologist who has the task of giving the distribution of pre- 
cipitation over a region should be aware of these inequalities and 
discontinuities. 



CHAPTER II 

GENERAL STATISTICAL CHARACTERISTICS OF 
CLIMATOLOGICAL ELEMENTS 

THE INTERNATIONAL MODEL of the meteorological (climato 
logical) register, covering the period of a month, contains 
about 750 observations; 9,000 observations for the year. 
The registers of a small climatological network with one hundred 
stations yield 900,000 observations in one year. The 5,500 co- 
operative stations of the U. S. Weather Bureau produce registers 
with about 12 million entries per year. The human brain is not 
capable of comprehending such a huge mass of numbers. Thus, 
a statistical treatment of observational data is necessary. 

II. 1. THE SYSTEM OF CHARACTERISTICS 

The following pages present a selection of such methods of 
mathematical statistics as are suitable for abstracting the common 
characteristics of climatological elements. L. W. Pollak has made 
a simple survey of these characteristics, and his system is based on 
the amount of mathematical operations necessary to compute them. 
They are divided into groups: l 

1) Primitive characteristics 

2) Elementary characteristics 

3) Higher characteristics. 

II. 2. PRIMITIVE CHARACTERISTICS 

Primitive characteristics can be derived from the original ob- 
servations by direct copying, by grouping under special headings, 
by counting, and by similar primitive methods. Examples will 
be given to pave the way to better understanding and to illustrate 
the close connection with climatological topics. 

II. 3. THE VARIATE. THE ABSOLUTE EXTREMES. 
THE RANGE OF VARIATION 

An example is the January minimum temperature at Mount 
Washington, N. H., for the four years 1938 to 1941. 

1 L. W. Pollak, Charakteristiken der Luftdruck-Frequenz-Kurven und verallgemein- 
erte Isobaren, Prager Geophysikalische Studien, vol. I (Prague, 1927). 

17 



18 METHODS IN CLIMATOLOGY 

First, a list of the data, as in Table 1, is required. In this list, 
the extremes of the minima are easily checked for each year. It is 
practicable to italicize the maxima, and to mark the minima by 
asterisks. Then the extremes of the entire sample are seen at first 



TABLE 1. DAILY MINIMA OF TEMPERATURE (F) AT MOUNT WASHINGTON, 
N. H., FOR FOUR MONTHS OF JANUARY 

(Mount Washington Observatory News Bulletin) 



January 1938 1939 1940 1941 



1 


2 


Q 


-14 


6 


2 


5 


-10 


-14 


9 


3 


1 


-21 


-10 


9 


4 


8 


2 


-11 


6 


5 





16 


- 3 


- 2 


6 


1 


19 


-14 


- 2 


7 


15 


20 


- 6 


1 


8 


- 3 


- 2 








9 


-13 








6 


10 


-10 


16 





- 3 


11 


-10 


2 





7 


12 


1 


-13 


5 


- 2 


13 


4 


-12 


11 


-30* 


14 


- 4 


- 4 


8 


-30* 


15 


4 


- 8 


4 


-24 


16 


- 5 


6 


-30* 


- 1 


17 


- 5 


-15 


-30* 


19 


18 


-15* 


-15 


-28 


21 


19 


- 2 





-16 


-12 


20 


2 


- 4 


-15 


-12 


21 


4 


- 4 


-11 


- 6 


22 


8 


2 


-19 


12 


23 


9 


-36* 


- 8 


- 6 


24 


11 


-10 





4. 


25 


19 


-22 


- 9 


3 


26 


- 5 


-29 


-14 


- 5 


27 


-13 


-13 


-12 


6 


28 


-11 


-12 


-13 


5 


29 


-13 


8 


- 8 


-12 


30 


- 4 


4 


- 8 


- 1 


31 


- 6 


- 1 


- 5 


- 4 



m - 1.1 - 4.7 - 8.7 - 1.5 -4.0 



glance. These are the absolute extremes of the sample of the variate, 
that is the variable quantity. In Table 1, the variate is repre- 
sented by the daily minimum temperatures for the months of 



GENERAL STATISTICAL CHARACTERISTICS 19 

January. We get : 

Absolute maximum = 21F 
Absolute minimum = 36F 



Difference = 57 F 

This difference is called the range of variation, and defines the 
entire range of the sample. The range is divided into a number 
of equal class intervals. 

II. 4. CLASS INTERVALS. FREQUENCY DISTRIBUTION. 
ABSOLUTE AND RELATIVE FREQUENCY 

The choice of the size of the class intervals is dependent upon 
the number of observations included in the sample and on the 
degree of exactness required. For the conditions of the model in 
Table 1, 4F-intervals are more or less suitable, because the sample 
consists of only 124 elements. 

TABLE 2. FREQUENCY TABLE. GROUPING THE DATA INTO 
CLASS INTERVALS OF 4 F 

Class Frequency 



Number (*) 


Limits ( 


[F) 


Center 


Absolute 


Percentage (%) 


(1) 


(2) 




(3) 


(4) 


(5) 


1 


-36 to 


-33 


-34.5 


1 


0.8 


2 


.... -32 to 


-29 


-30.5 


5 


4.0 


3 


.... -28 to 


-25 


-26.5 


1 


0.8 


4 


.... -24 to 


-21 


-22.5 


3 


2.4 


5 


- 20 to 


-17 


-18.5 


1 


0.8 


6 


.... - 16 to 


-13 


-14.5 


15 


12.1 


7 


-12 to 


- 9 


-10.5 


16 


12.9 


8 


- 8 to 


- 5 


- 6.5 


13 


10.5 


9 


.... 4 to 


- 1 


- 2.5 


19 


15.3 


10 


to 


3 


1.5 


19 


15.4 


11 


4 to 


7 


5.5 


13 


10.5 


12 


8 to 


11 


9.5 


9 


7.3 


13 


12 to 


15 


13.5 


2 


1.6 


14 


16 to 


19 


17.5 


5 


4.0 


15 


20 to 


23 


21.5 


2 


1.6 



* For some statistical purposes it is more desirable to begin with no. (zero). See: 
E. Czuber, Die statistischen Forschungsmethoden (Wien, 1921), p. 91; and L. W. Pollak, 
Prager Geophysikalische Studien, vol. I, pp. 33 and 34. 

These class intervals are seen in the second column of Table 2. 
In many cases it is more advantageous to give the central height 
or midpoint of the class interval, as in column 3. The first column 
contains the serial numbers of the class intervals. 



20 METHODS IN CLIMATOLOGY 

The number of observations falling within the limits of the 
consecutive intervals is counted. The resulting series represents 
the frequency distribution of the sample in question. Column 4 in 
Table 2 gives the frequency distribution of minimum temperatures, 
etc. These numbers are called absolute frequencies. For funda- 
mental climatological purposes of comparison, relative frequencies 
are more expedient ; these are absolute frequencies expressed in per 
cent of the total number (n) of observations. (In the example of 
Table 2, n = 124.) Examples of the great usefulness of frequency 
distributions, which often mean an indispensable supplement to the 
usual averages, can easily be given. 

The figures in Table 3 are taken from a war climatology (World 
War I), when it was necessary to give an effective comparison of 
summer temperatures in Central Europe (Vienna, Austria: 48.2N, 
16.3E, 660 feet) and the southern Adriatic coast (Scutari: 42.1 N, 
19.4E, 70 feet). One might think that a simple comparison of 
temperature maxima would be illustrative. In reality, in Scutari, 
a maximum of 98F has been observed within a period of more than 
30 years, in Vienna 98F was also observed, though within a period 
of 70 years. The maxima are practically identical. 

A frequency distribution of terryperature in Vienna and Scutari, 
however, shows immediately the characteristic and great climatic 
differences of the two places in spite of the fact that the comparison 
considers the temperatures only at 7 A.M. of the summer months of 
a synchronous period of 7 years. Each of the samples in Table 3 
comprises 92 X 7 = 644 observations. 

TABLE 3. DISTRIBUTION OF RELATIVE FREQUENCIES (%) OF TEMPERATURE AT 

7 A.M. AT A PLACE IN CENTRAL EUROPE (V) AND AT A PLACE ON 

THE SOUTHERN ADRIATIC COAST (S) IN SUMMER 

(JUNE, JULY, AUGUST) 

Class interval V S 

F % % 



41.0 to 49 .9 


0.5 


_ 


50.0 to 58.9 


35.4 





59.0 to 67.9 


59.2 


19.9 


68.0 to 76.9 


4.9 


703 


77.0 to 85.9. . 




98 


S 


100.0 


100.0 









From Table 3 it is seen that at 7 A.M., after the cooling during 
the night, 95% of all observations are below 68F in Central 
Europe; on the Adriatic coast 80% exceed 68F. At first glance, 



GENERAL STATISTICAL CHARACTERISTICS 21 

it is also seen that the distribution is less scattered in Scutari 
(3 classes) and more in Vienna (4 classes). 

These facts reveal a drastic contrast of temperature conditions 
in the two regions. It is not the absolute height of the maxima 
which counts. It is, rather, the fact that the greatest frequency 
is shifted to the next higher class on the Adriatic Sea. That means 
great frequency of high temperatures during the night and in the 
morning hours, a condition enervating for people accustomed to a 
more moderate climate. The frequency distribution points to 
the greater monotony of high temperatures in the regions of the 
Adriatic Sea during the summer. 

The foregoing example shows the value of relative frequencies 
for purposes of comparison. Besides, the size of the class interval 
is notable. With the sample in Table 3 the size of the interval is 
not less than 9 F, sufficient for the desired purpose. 

II. 5. HISTOGRAM AND FREQUENCY CURVE. 
THE MODE. THE MODAL CLASS 

As a consequence of the relatively small intervals in Table 2, 
the distribution takes a rather irregular shape, which is well illus- 
trated by the histogram of Figure 4. This figure contains two 
methods of representation: the histogram, " rectangular areas 
standing on each grouping interval showing the frequency of 
observations in that interval," 2 and the frequency curve which 
has been smoothed. In reality, the variation in question is dis- 
continuous, because the variate is confined to whole degrees 
Fahrenheit (see Table 2). The frequency curve is more usual in 
climatological statistics, and there is no real reason "for insisting 
on the histogram form." The latter may be more appropriate from 
a theoretical standpoint. The continuous curve gives a better 
survey, and the maximum of the curve indicates the approximate 
value of the mode, the most frequent value. This procedure is 
based on the assumption that a large number of observations are 
available to give a smooth distribution of frequencies. In the 
present example, that is not realized. 

2 R. A. Fisher, Statistical Methods for Research Workers (6th ed., London, 1936), p.38. 
The frequencies can be plotted on the ordinates through the midpoints of the class in- 
tervals. "The tops of all the ordinates are joined by a broken line, 11 i.e. the original 
frequency curve which appears as a smoothed line in Figure 4. See D. Brunt, The Combi- 
nation of Observations (Cambridge, England, 1931), p. 7. 



22 



METHODS IN CLIMATOLOGY 



Another important concept is the modal class, that is, the class 
interval into which the greatest number of observations fall. 

In the example of Table 2, there are two equal modal class 
intervals, nos. 9 and 10. This offers a certain difficulty in 




O> 10 K *> 0> 
fOCSICMM -- 
ii i | I ' l 



IO t- 



lO <J> lO 
04 



8 - 



FIG. 4. Histogram of relative frequencies. A smoothed frequency curve 
is added. Minimum temperatures of four months of January at Mount Wash- 
ington, N. H. 

puting the temperature which represents the mode. Even in this 
case, histogram and frequency curve permit an estimate of the 
mode. In the example in question, the mode = ( 2.5 + l.S)/2 
= - 0.5F. 

Generally, the value of the mode should be approximated by a 
simple linear interpolation ; the equation reads : 8 



Mode = L mo + 



C 



H. Arkin, R. R. Colton, An Outline of Statistical Methods (4th cd. f New York, 1939), 
p. 23. 



GENERAL STATISTICAL CHARACTERISTICS 23 

Where 

L mo = Lower limit of modal class 
f a = Frequency of class interval above modal group 
/6 = Frequency of class interval below modal group 
C = Size of class interval 

As stated before, in the example of Table 2 the class intervals 
nos. 9 and 10 include equal numbers of frequency. Therefore 
they are combined into one modal class with a frequency num- 
ber = 38. 

In order to avoid unequal intervals, nos. 11 and 12 are taken 
collectively as/ a and nos. 7 and 8 as/6. 

Thus we get from Table 2 : 

Lmo = 4F (lower limit of interval no. 9) 
f a = 13 + 9 = 22 (nos. 11 and 12) 
/ 6 = 13 + 16 = 29 (nos. 8 and 7) 

C = 8F 



Therefore : 



Mode = - 4 + 22^29 8 = ~ aS5 F 



Accidentally, the two results are in perfect agreement. 

Generally, frequency distributions are necessary and indis- 
pensable in climatological investigations. This does not hold to 
the same extent where mode is considered. Variates taken from 
the realm of climatology, in particular, show remarkable frequency 
curves with two or more modes, so that the mode can no longer be 
considered as an unequivocal characteristic. Frequency curves 
of winter temperatures, for instance, show this phenomenon, 
especially at places located in a transition zone between an oceanic 
climate and a strongly continental one. 

Two modes exhibit apparently just about the same frequency: 
one, at a temperature of some degrees below the freezing point, 
corresponding to the average temperature of fresh continental air 
masses (Leningrad, 59.9N, 30.3E: about 20F); the other mode, 
at about the freezing point. This means that the heat content of 
the advected oceanic air masses at higher temperatures is expended 
in the process of melting the snow-cover, its thermostatic effect. 

We learn from this example that, in the great number of similar 
cases, two (and even more) modes can exist, so that the bulk of 



24 METHODS IN CLIMATOLOGY 

observations cannot be characterized by one number alone, in 
contradistinction to the arithmetical mean. In other respects, 
also, the mode of climatic series happens to have properties which 
do not make it a desirable statistical element. On the other hand, 
one sees from the last example that the double mode of the tem- 
perature at a place (like Leningrad on the Baltic Sea) leads to a 
clear physical analysis of an important climatic feature. Even 
in this respect, it should not be forgotten that the frequency curve 
conveys the same knowledge, and moreover a comprehensive 
view of all observations. 

II. 6. ELEMENTARY CHARACTERISTICS. 
MEDIAN, QUARTILES, DECILES 

The method of computing the median stands between the 
primitive and the elementary characteristics, because there are at 
least theoretically two methods, the first of which leads immedi- 
ately to a primitive element. 

All items of which the sample consists are arranged according 
to size. This series is called the array. If the number of these 
items is odd, then the middle valu^ is called the median. If the 
number is even, then the average of the two central items is de- 
fined as the median. This method obviously represents the 
median as a primitive characteristic. 

Example: Take the series of minimum temperatures in January 
1938 from Table 1 and arrange them according to magnitude. The 
series of Table 4 A results. The median = 2F. If the last 

TABLE 4. A. EXAMPLE FOR AN ARRAY. (TEMPERATURE MINIMA 
JANUARY 1938, FROM TABLE 1) 

No. 1 2 3 4 5 6 7 8 9 10 11 

-15 -13 -13 -13 -11 -10 -10 -6 -5 -5 -5 

No. 12 13 14 15 16 (median) 17 18 19 20 

-4 -4-4-3-20111 

No. 21 22 23 24 25 26 27 28 29 30 31 

2 2 4 4 5 8 8 9 11 15 19 

B. GROUPED DATA 

4 

No. F Frequency SF< 

1 -15 to -7 7 7 

2 - 6 to 2 15 22 

3 3 to 11 7 29 

4 12 to 20 2 31 



GENERAL STATISTICAL CHARACTERISTICS 25 

item (no. 31 = 19) were omitted so that the series had an even 
number of items = 30, the median would be the average of the 
two central items, nos. 15 and 16: 



- - 2.5F 



This method of arranging is " primitive, M but as a rule very tedious, 
because of the magnitude of climatological samples. 

The second method must be applied to samples which are al- 
ready grouped into class intervals. For this purpose, a sample with 
a greater number of items is necessary. Table 5 contains a sample 

TABLE 5. FREQUENCY DISTRIBUTION OF MINIMUM 
TEMPERATURE (C) OF 25 MONTHS OF JULY* 

(Coordinates for the place K: 48.1N t 14.1E, 1280 feet) 



0) (2) 

No. (0 Central height 
C 


(3) 
Ft 


(4) 

-u 

2 Fi 
-i 


1 6.45 


1 


1 


2 7.45 


1 


2 


3 8.45 


4 


6 


4 9.45 


15 


21 


5 10.45 


52 


73 


6 11.45 


84 


157 


7 12.45 


131 


288 


8 13.45 


121 


409 


9 14.45 


108 


517 


10 15.45 


114 


631 


11 16.45 


75 


706 


12 17.45 


45 


751 


13 18.45 


13 


764 


14 19.45 


9 


773 


15 20.45 


2 


775 









* Numbers of Table 5 taken from F. Steinhauser, Meteorologische Zeitschrift, 1935, 
p. 207. 

of daily minimum temperatures (C) for 25 months (of July at a 
place in the foothills of the eastern Alps). 

The absolute extremes of this sample are: 6.3C and 20.9C 

The range of variation is: 14.6 C 

The number of items is: 775 

As this is a sample of greater magnitude with an exactness of 



26 METHODS IN CLIMATOLOGY 

0.1 C of the individual observations, 1 C is chosen as a suitable 
size for the class intervals. 

The first interval reaches from 6.0C to 6.9C 
the second from 7.0C to 7.9C 
the last from 20.0C to 20.9C 

The central heights of the class intervals are therefore : 



The first column of Table 5 contains the number of the class 
interval; the second, the central height of the class interval; the 
third, the frequency within the class intervals; and the fourth, the 
series of cumulated frequencies. 

i=15 

This series ^i is calculated as follows : The frequency of the 
t=i 

interval no. 1 = 1; the frequency of nos. 1 and 2 = 1 + 1 = 2 etc. 
The last sum (at no. IS) represents therefore the whole size of the 
sample, i.e., 775. 4 

The median value is defined in such a way that 50% of the 
sample lies below it and 50% above. The whole size is: 

N = 775 

50% otN: f = 387.5 
// 

From column 4, it is seen that this limit is surpassed within the 
8th class interval. Therefore the median is located between 1 3 .0C 
and 13.9C. In this case, a simple linear interpolation leads also 
to the result. 

The 8th interval contains 121 items, of which 99.5 (387.5 - 288) 
should lie below the median value. Consequently, the interval of 
1.0C (if we start from 12.9C) has to be divided in the propor- 

4 S, the capital Greek S, means the sum of a given number of consecutive elements 
of a series. 
b 
2 means that the sum is to be taken from the element with the number (subscript) 

a 

a to an element no. b. 



GENERAL STATISTICAL CHARACTERISTICS 27 

tion of 

99 5 

- - 82 



so that the median is at 12.9C + 1.0 X 0.82 = 13.72C. 6 

According to the same principle, the entire size of the sample 
can be divided into an arbitrary number of equal parts, which are 
symmetrical about the median. It is usual to divide the sample 
into 10, or into 4, parts. The limits of the first are called deciles, 
those of the second, quar tiles. 

In the example of Table 5, the limits of the deciles are given by 
the cumulative sums of N/W = 77.5, with the following results: 

Decile no. 12345 

Upper limit number of items 77.5 155.0 232.5 310.0 387.5 

Decile no. 6 7 8 9 10 

Upper limit number of items 465.0 542.5 620.0 697.5 775.0 

The procedure of computation is now identical with that used 
for the median. The first decile contains 77.5 items, and therefore 
it lies within the 6th class interval between 11.0C and 11.9C. 
Because the first 5 class intervals contain 73 items, the 6th class 
interval participates with 4.5 items to give the first decile. There- 
fore, the entire size of the 6th class interval has to be divided in 
the proportion of 

g - <* 

The upper limit of the first decile is consequently at: 
10.90 + 1.0 X 0.05 = 10.95C 

Herewith it is assumed that the sample is composed of discrete 
numbers, e.g., 6.0, 6.1, etc. 

6 Part B of Table 4 shows the data of part A grouped into class intervals of 9F. 
There 

-15.5 

Because the first interval contains 7 items, 8.5 items (15.5 7 = 8.5) of the second 
interval lie below the median. Therefore this interval of 9 F with 15 items must be 
divided in the proportion of 



and the median lies at: - 7 + 9 X 0.57 = - 2F. It is worth noting how good the 
agreement is between the method of an array (see Table 4A: 2F) and the method in 
question, even for a sample of only 31 elements. 



28 METHODS IN CLIMATOLOGY 

Reduced to one decimal place, we get 

1LOC 

The second and first deciles together contain 155.0 items. 
The second decile also lies within the 6th class interval which 
participates with 82 items. The class interval must be divided 
in the proportion of: 



And the upper temperature of the second decile lies at : 

10.9 + 1.0 X 1.0 = 11.9Cetc. 
Thus the following division of the sample results : 

C 

Absolute minimum 6.3 

Decile 1 11.0 

Decile 2 11.9 

Decile 3 12.5 

Decile 4 13.1 

Decile 5 * 13.7 = median 

Decile 6 14.4 

Decile 7 15.1 

Decile 8 15.8 

Decile 9 16.8 

Absolute maximum 20.9 

The deciles divide the sample into equal parts of 10%; the 
quartiles into parts of 25%. As stated above, there is no other 
difference in principle between them. The computation of the 
quartiles needs no further explanation. 

In the example of Table 5, N = 775, the cumulative sums for 
the quartiles would be 193.75; 387.50; 581.25; 775.00. Therefore: 

Absolute Lower Upper Absolute 

minimum quartile Median quartile maximum 

6.3 12.2 13.7 15.5 20.9C 

5.9 C 1.5C 1.8C 5.4 C 

It is characteristic of the element under investigation that 50% 
of all observations lies within 3.3 C (between 12.2C and 1S.SC). 
The other 50% is distributed over a range of 11.3C altogether. 
These facts give a desirable picture of the rarity of greater devia- 
tions from the mean value. 



GENERAL STATISTICAL CHARACTERISTICS 29 

The size of the interval between upper and lower quartiles is 
always of interest because 50% of the entire variate falls within 
these limits and 50% beyond them. There is the equal likelihood 
that an observation falls inside or outside of the limits of the 
quartiles. The small range between the lower and upper quartiles 
shows that the density of the items is much greater inside this 
space than outside. 

The three characteristics of the frequency curve median, 
quarries, deciles have been discussed, and a simple method has 
been shown of computing these statistical values. This has been 
necessary because in the English meteorological and climatological 
literature these characteristics are frequently used. Furthermore, 
these methods assume their real value only if the size of the sample 
is so large that half the number of its items represents a series of 
many hundreds. If, then, the comparison of the characteristics in 
question showed significant differences between two or more 
periods of years, we should infer "that the climate (or the methods 
of measuring it) had materially altered." 6 

On the other hand, incontestable climatic series of great length 
large samples are fare; with small samples, the results from 
these characteristics are often disappointing. Further discussion 
of this subject is therefore superfluous. The frequency curve it- 
self is undoubtedly one of the best and finest means in the hand of 
the climatologist. 

R. A. Fisher, Statistical Methods, p. 42. 



CHAPTER III 

SPECIAL STATISTICAL CHARACTERISTICS OF 
CLIMATOLOGICAL SERIES 

III. 1. THE ARITHMETICAL MEAN 

IT GOES without saying that the arithmetical mean is defined 
by: 

a = a 

where a denotes the arithmetical mean, a,i represents any number 
from a\ to a n and n is the number of observations which constitute 
the series. 

The computation of the arithmetical mean even of large series 
is simple, especially if a calculating machine is at hand. If the size 
of the sample is extraordinarily large, there are methods for com- 
puting the arithmetical mean from* the sample divided into class 
intervals, when the midpoints and the frequencies are known. 
For these, the reader is referred to textbooks of statistical methods, 
since such cases are not sufficiently significant for climatology. 

The use of the arithmetical mean (average) in climatology is 
common and need not be fully discussed. Here, however, a few 
remarks about averages of extremes may be opportune, because of 
frequent misunderstandings. The following examples are taken 
from Table 1, minimum January temperatures on Mount Washing- 
ton, New Hampshire. 

(a) Mean daily extremes; in this case "minima" 
1938: -1.1; 1939: -4.7; 1940: -8.7; 1941: -1.5F 

(b) The x-years' average of the mean daily extremes ; in this 
case the four-years' average of the mean daily min- 
ima = - 4F 

(c) Average absolute extremes calculated from the absolute 
extremes of each year (month etc.) ; in this case 
lowest minima: (-15, -36, -30, -30): 4 = - 28F 
highest minima: ( + 19, +20, +11, +21):4 = + 18F 

(d) The x-years' average range of the extremes, in this case 
of the minima, is 46 F 

30 



SPECIAL STATISTICAL CHARACTERISTICS 31 

The examples are taken arbitrarily from the topic of temperature 
in order to dispense with additional tables ; the underlying concepts 
are applicable to any element of scalar character. 

III. 2. DEVIATIONS. AVERAGE VARIABILITY. 
STANDARD DEVIATION. NORMAL DISTRIBUTION 

There may be a series of n numbers representing the results of n 
observations : 

0i, #2, #n 

The arithmetical mean is: 




Then the differences between the single elements of the series a f - 
and the arithmetical mean, i.e., 

di = a\ a; d% = a% a d n = a n a 

are called deviations, which play the greatest role in the representa- 
tion of statistical and also of climatological series of numbers. 

The sum of the deviations from the arithmetical mean is zero, if 
the signs of the deviations are considered. 1 

The sum of the deviations, without regard to sign, divided by 
their number is called mean deviation (/z) in statistical mathematics 
and average variability (AV) in climatology. Therefore : 






n 
Here: S| | means the sum of the absolute values, ignoring signs: 

M = (\di\ + \d*\ + + \d n \):n 

Table 6 gives examples for the computation of deviations and 
of the average variability (mean deviation). The dates of the 
last killing frost in Vancouver, B.C., and in Bismarck, N. D., in the 

1 In American papers and books the expression "deviation" is frequently replaced 
by the term "departure" and "mean deviation" by "mean departure." The term 
"departure" is seemingly not so common in British investigations and is even lacking in 
the Meteorological Glossary (London, 1939), and in Shaw's Manual of Meteorology. 
The term "average variability" is usual in numerous climatological papers of many 
countries and has its advantages from the didactic standpoint. 



32 METHODS IN CLIMATOLOGY 

TABLE 6. DATES OF THE LAST KILLING FROST IN VANCOUVER, B.C., AND IN 
BISMARCK, N.D., AND DEVIATIONS (d) FROM THE MEAN DATE (rft) 



Year 


Vancouver, B. C. 
Date Year day* 
(D (2) 


d 

(3) 


d* 

(4) 


Date 
(5) 


Bismarck, 
Year day* 
(6) 


N. D. 
d 
(7) 


<** 

(8) 


1901 


IV 24 


114 


+ 6 


36 


VI 7 


158 


+24 


576 


2 


12 


102 


- 6 


36 


IV 29 


119 


-15 


225 


3 


15 


105 


- 3 


9 


V 5 


125 


- 9 


81 


4 


23 


113 


+ 5 


25 


14 


134 








5 


III 31 


90 


-18 


324 


12 


132 


- 2 


4 


6 


IV 12 


102 


- 6 


36 


27 


147 


. +13 


169 


7 


29 


119 


+ 11 


121 


14 


134 








8 


28 


118 


+ 10 


100 


2 


122 


-12 


144 


9 


21 


111 


+ 3 


9 


13 


133 


- 1 


1 


10 


14 


104 


- 4 


16 


17 


137 


+ 3 


9 








+35 








+40 




2 




1078 


-37 


712 




1341 


-39 


1209 








72 








79 




in 


IV 18 


108 


7.2 




V 14 


134 


7.9 




a 








8.44 








11.00 



* Number of the day from first of January. 

The data are taken from Frank J. Bavendick, "Climate of North Dakota," p. 1048, 
and Lawrence C. Fisher, " Climate of Washington, " p. 1175, Climate and Man, Yearbook 
of Agriculture (Washington, D. C., 1941). 

ten years 1901 to 1910 are given. For computation, the dates 
(columns 1 and 5) have to be converted to numbers of days starting 
January 1. (February 1 = no. 32, etc.; see below, Appendix IV). 
These numbers are shown in columns 2 and 6. The average dates 
(arithmetical means) are the 108th and 134th days of the year, or 
April 18 and May 14 respectively. In columns 3 and 7, the devia- 
tions are shown and their sum total (disregarding the signs) is 
given. The sums of the positive and of the negative deviations 
should be identical (small differences being due to rounding off the 
arithmetical mean (rri) to integers). 2 

The sum of the absolute values of the deviations divided by 
their number (10) equals the average variability, which represents 
one of the most valuable climatological elements derived from the 
observations. From the examples, the variability of the date of the 
last killing frost is somewhat greater in the interior of the continent 
than at the coast. The principal contrast is given by the differ- 

2 The differences should be checked. In the case of column 2, the exact rh = 107.8. 
Each of the 10 deviations is afflicted therefore with an error of 0.2. Hence, the differ- 
ence 10 X 0.2 = 2, in agreement with the result in column 3. In column 6 the exact 
mean is 134.1 instead of 134. The difference is 0.1 X 10 1. 



SPECIAL STATISTICAL CHARACTERISTICS 33 

ence in the average dates. Inland, the average last killing frost 
occurs 26 days later than at the coast. 

A rough measure of the variability is also given by the difference 
between the absolute extremes, i.e., 29 days at Vancouver and 39 
days at Bismarck, N. D. This measure is called "rough" because 
it depends upon only single values. For instance, if the series 
began with 1902, the differences would have been 29 days (un- 
changed) and 28 days, respectively. The average variability 
considers each value of the series, and is the usual method of 
characterizing the scatter of the element in question. 

The second method is to compute the standard deviation 8 (a) 



cr = 




which is the best and most exact measure of scattering. In 
climatological investigations it is not so frequently used, since the 
calculation is more complicated. This problem will be treated 
later. 

The example of Table 6 gives the following results (columns 
4 and 8): 

= 712 2<Z 2 * = 1209 





The standard deviation, the scatter of the date of the last killing 
frost, is about 30% greater inland than at the coast. 

From the frequency distribution shown in Figure 4 and Tables 
4B and 5, one draws the simple and natural conclusion that ex- 
treme values are rarer than values close to or within the modal 
class interval. Under certain conditions, this experience holds, 
too, in the case of deviations if the inferences are logically changed. 
This means that the smaller the deviations, the more frequent 
they are. 

The ideal case is realized by the so-called normal distribution, 
or Gaussian distribution, which governs deviations independent 
of one another. 

This is the " standard deviation" of the single value; that of the arithmetical 
mean is: 

' _L 
* Vn 



34 METHODS IN CLIMATOLOGY 

There should be no effective reason why the deviations, if their 
number be large, may not take any value from <*> to + 
Then, the frequency curve is a bell-shaped curve which fulfills the 
equation : 



y = 



where y means the frequency ordinate at the distance d (deviation) 
from the center of the curve (Fig. 5), TT = 3.142 (Ludolf s number), 




FIG. 5. Normal distribution showing the areas (per cent) between ordi- 
nates in distances of one, two, three Standard Deviations left and right from 
the center (frequency of the arithmetical mean) 

e = 2.718 (base of the natural logarithms). This is the law of 
Karl Friedrich Gauss. 4 

Obviously, the curve represented by this equation is sym- 
metrical, with the greatest frequency at the center, because equal 
values of y are related to ( d) and to (+d). The greatest fre- 



4 For a more correct representation, L. H. C. Tippett (The Methods of Statistics, 
London, 2nd ed. f p. 54) may be quoted: "The frequency curve is deduced from a histo- 
gram and consequently areas under the curve, not heights of the ordinates, represent 
frequencies. It is therefore appropriate to use the notation and ideas of the integral 
calculus, to imagine about any given value of x an elemental sub-range dx, and to regard 
the area under the curve between ordinates drawn at the limits of the sub-range as an 
element of frequency, df. Then this elemental strip may be regarded as a rectangle of 

N 
height y and df = = e^^'dx" where N denotes the number of items. 



SPECIAL STATISTICAL CHARACTERISTICS 35 

quency occurs at d = 0, if 



Therefore the arithmetical mean is the most frequent value, and 
coincides with the values of the median and the mode. This is 
valid only if there is a symmetrical distribution of frequency of 
the deviations with an absolute maximum at d = 0. 

Gauss's equation shows that a normal frequency distribution 
is fully defined by its respective standard deviation <r. 

Geometrically, <r is the distance on either side of the center 
where the slope is steepest, at the points of inflexion of the curve. 
(See Fig. 5.) 

In practical applications we are not so much interested in the 
frequency at any distance from the center. It is much more im- 
portant to know the total frequency (or probability) beyond this 
point (for instance beyond the distance a in Fig. 5). This fre- 
quency or probability is represented by the area included between 
the ordinate at the chosen point, the tail of the curve, and the 
X-axis. 

For deviations expressed in units of cr, (d/d), tables of this 
total frequency, or probability integral, have been constructed, 
from which, for any value d/cr, we can find what fraction of the 
total sample has larger deviations. 

III. 3. A PRACTICAL APPLICATION OF THE 
STANDARD DEVIATION IN CLIMATOLOGY 

The probability (P) 5 that a deviation exceeds 1, 2, or 3 times 
the standard deviation is 0.317; 0.047; 0.003. The numbers 
illustrate the rapid decrease of the probability of increasing 
deviations. 

In other words, the area between the ordinates at: cr (Fig. 5) 
2cr, 3cr includes: 

between - or and -f a: 68.260% 
between -la and +2cr: 95.346% 
between -3<r and +3cr: 99.729% 
between -4(7 and +4cr: 99.994% 

1 That is the ratio of the number of favorable events to the total number of possible 
events. If a coin is tossed, only " heads" or "tails" can appear. The probability of 
heads (e.g.) is therefore 1/2. 



36 METHODS IN CLIMATOLOGY 

In climatological investigations, the probable error (f) can be 
useful because it includes exactly 50% of the sample between +/ 
and /. The probability that a deviation is </ is therefore 1/2. 

The distance of the ordinate from the center which borders this 
area of the curve is called quartile distance^ a term the meaning of 
which is clear from previous explanations. The percentages of 
the areas included between the ordinates at /, 2/, etc. are: 



between + / and - /: 50.00% 
between +2/and -2/: 82.26% 
between + 3/and -3/: 95.70% 
between +4/and -4/: 99.30% 



The probable error is usually computed from the standard 
deviation according to the relation : 

/ = 0.6745<r 

Example of the practical use of the division of the area 
of the curve of normal distribution 

Often the climatologist has to determine the rareness of an 
actual observation of temperature, precipitation, etc., or else of a 
mean intensity, or amount, such as a mean of temperature, sum 
of precipitation of an individual month, etc. Statements to that 
effect are necessary on the occasion of an expert opinion and in the 
description of a climate. Questions may arise such as: Is a wind 
velocity of x mi/hour, is a temperature of x degrees below zero, is a 
downpour of x inches in an hour, etc., a normal event in a given 
climatic region, or is the observed value rare, or very rare? The 
courts are especially interested in such estimates because in the 
latter case the judge can declare the event to be a force majeure 
(superior force). For many other purposes, including military 
ones, it is highly important to know the limits between which 
normal values of the climatic elements can vary. 

Usually such questions are answered by the climatologist ac- 
cording to common sense. 

The division of the area of the curve of normal frequency 
distribution provides a transition from guessing to quantitative 
estimation. If a long series of observations of the element in 
question is at hand, one can calculate cr or/ (standard deviation or 
probable error). 



SPECIAL STATISTICAL CHARACTERISTICS 37 

Then the following nomenclature can be used : 

below: 3<r( 3/): extremely subnormal = ES 

between: 3o-( 3/) and 2<r( 2/): greatly subnormal 

2o-( 2/) and <r( /): subnormal = 5 

~ *( f) and + cr(+ /): normal = J 

+ <jr( + /) and +2cr( + 2/): above normal = A 

+ 2cr( + 2/) and + 3 cr ( + 3/) : greatly above normal = 
above: +3o-( + 3/) extremely above normal = JS-4 

At State College, Pennsylvania, 6 53 years' observations of the 
mean daily minimum of the coldest month were available. The 
probable error is:/ = 2.9 F. 

Hence the following classification (as indicated in Table 7). 
The class limits are derived from the 53 years 1886 to 1938. It 
should be kept in mind that these limits are changed if another 
period is used. On the other hand, the variability of the limit is 
smaller by far than that of the values of the element itself. In 
climatographies, such classifications of temperature, precipitation, 
etc., should be given for each place with about 50 or more years' 
observations. 

From Table 7, one concludes that a deviation from the mean 
daily minimum of > 8.7F is to be considered a rare event, with 
a theoretical probability of two occurrences in 100 years. Conse- 
quences from such low temperatures (ruin of winter wheat, of 
rosebushes, etc.) could be judged as caused by a " superior force/ 1 

TABLE 7. CLASSIFICATION OF DEVIATIONS FROM THE MEAN DAILY MINIMUM 
TEMPERATURE AT STATE COLLEGE, PA., BASED ON THE PROBABLE ERROR 



(1) 


(2) 


(3) 


(4) 


(5) 


Charac- 




Observed 


Calculated 




teristics 


Classification 


frequency 


frequency 


Obs. - Calc.* 


ES 


< - 8.7 F 


1.88 


1.80 


+0.08 


GS 


-8.7 to -5.8 


7.55 


6.72 


+0.83 


S 


-5.8 to -2.9 


18.87 


16.13 


+2.74 


N 


-2.9 to +2.9 


45.28 


50.00 


-4.72 


A 


+ 2.9 to +5.8 


18.87 


16.13 


+2.74 


GA 


+ 5.8 to +8.7 


7.55 


6.72 


+0.83 


EA 


> +8.7 


0.00 


1.80 


-1.80 



* Column 3 minus column 4. 

8 V. Conrad, "Investigation into Periodicity of the Annual Range of Air Tempera- 
ture at State College, Pennsylvania," The Pennsylvania State College Studies, No. 8 
(State College, 1940), pp. 10 ff. See also F. H. Chapman, Meteorological Office, London: 
Professional Notes, 1919, no. 5; and V. Conrad, Meteorologische Zeitschrift, 1921, p. 91. 



38 



METHODS IN CLIMATOLOGY 



The example in Table 7 shows that the " probable error" has 
the advantage that the intervals are rather smaller than at the 
standard deviation. This is valuable for climatical purposes 
since limits of rare cases are better adjusted to common feeling. 

The theoretical frequency distribution is represented as a full 
line in Figure 6, and the crosses mark the observed frequencies at 
the midpoints of the class intervals. It should be pointed out that 



o/ 
7 

20 



10 




-10.2 -7.3 -4.4 -1.5 



1.5 4.4 



7.3 !0.2F 



FIG. 6. Frequency distribution (per cent) of average daily minimum 
temperatures in State College, Pa. 

Calculated : Full line. Observ%d : x.* (After V. Conrad) 

% 

* The numbers below the abscissa mean the midpoints of the class intervals. The 
normal class is divided into two class intervals. 

the observed frequency curve fits the normal distribution very well. 
The small departure within the modal class interval is called 
positive excess. 



III. 4. VARIANCE. COEFFICIENT OF VARIATION 

Before continuing the discussion of how far climatological series 
show a more or less normal frequency distribution, one may men- 
tion two further measures of the dispersion (scatter) of a variate. 
Both, e.g., are used in the papers and reports of the agricultural 
climatological department of British India. 7 They are directly 
derived from the standard deviation (o-). 

The first is the variance ( Va) : 



Va = 



n 



1 See, for instance, P. C. Mahalanobis, Report on Rainfall and Floods in North Bengal 
(Calcutta, 1927). 



SPECIAL STATISTICAL CHARACTERISTICS 39 

It is the square of the standard deviation, and is somewhat 
easier to calculate. The indications of the variance react a little 
more strongly to small changes of the distribution of deviations, 
as exemplified in the left side of Table 6 (Vancouver) : 



. 

n 10 

The other measure is the coefficient of variation (CV), that is, 
the standard deviation expressed as a percentage of the arithmetic 
mean (m) : 

CV= 100^=100 



if the numerical values are taken from Table 6 (Vancouver). 

The "coefficient of variation" should be more frequently used 
in climatological statistics; it permits an exact comparison of the 
values characterizing the dispersion because it is reduced to an 
equal arithmetical mean. 

III. 5. SUMMARY OF STATISTICAL CONCEPTS 

Thus far the following statistical expedients have been ex- 
plained : 

Frequency Distribution Mean Deviation (/*) 

Median Standard Deviation (<r) 

Quartiles Normal Distribution 

Deciles 8 Probable Error (/) 

Arithmetic Mean (in) Variance (Va) 

Deviations (d) Coefficient of Variation (CV} 

III. 6. RELATION BETWEEN /x, <r, AND /. CORNU'S THEOREM. 
UNILATERALLY AND BILATERALLY LIMITED VARIATES 

In the foregoing discussion only the relation of standard devia- 
tion (<r) and probable error (/) was mentioned : 

/ = 0.67450- 
Standard deviation (<r) and mean deviation (/z) are connected by 

8 To my knowledge, sextiles (6 parts) and percentiks (100 parts of a sample) are not 
usual in climatological investigations. For the rest, the method of calculation is iden- 
tical to that used for medians, quartiles, etc. 



40 METHODS IN CLIMATOLOGY 

the following simple equation : 

<r = A/^ ji = 1.253/i 

As a rough approximation one can use 

a = 5/4 M 

These relations are exactly valid only in the case of a normal 
distribution, and approximately valid if the departures from the 
normal distribution are not too great. If the size of the climato- 
logical sample is not too small (lower limit perhaps 25 items), a 
can be computed with the above equation if there is some likeli- 
hood that the variate can be represented more or less by a normal 
frequency distribution: An example may be taken from Table 6. 
The mean deviation /* (column 3) is: 

M = 7.2 
therefore 

, = 1.253/i = 9.02 

instead of 8.44. This means an error of 7% of the true value. 
The agreement mentioned is not too good, because the number of 
items is much too small. The equation 



a = 
can be written 

^ = TT = 3.1416 

This expression is called Cornu's Theorem and has often been 
used as a solution of the problem, whether or not one may assume 
that the climatological elements can be represented by a normal 
distribution. This inversion of the problem may be illustrated by 
the following examples. The deviations of 

125 mean winter temperatures at Vienna, Austria (Central 
Europe), yield 



- 3.H 

M 2 
125 mean summer temperatures at Vienna, Austria, yield 

- 3.25 



SPECIAL STATISTICAL CHARACTERISTICS 41 

130 average annual temperatures at Paris, France (Western 
Europe), yield 



= 3.16 

M 
110 average annual temperatures at Milan, Italy, yield 

^ = 3.11 

M 2 

130 average January air pressures at Paris yield 

^-3.13 

M 2 

From the ten years' data of the last killing frost in Vancouver 
(Table 6) it follows that: 

2<r 2 X 8.44* 



The differences between the other results are only 4% of TT. 

One might think that in general the series of deviations of a 
climatological element shows the normal distribution. An ex- 
ample of this behavior is given in Table 7 and Figure 6. On the 
other hand, it must be emphasized that the fulfilling of Cornu's 
theorem is only a necessary, but not a sufficient condition of normal 
distribution. This means that if Cornu's criterion is not satisfied, 
no normal distribution can be expected; but the criterion can be 
fulfilled even if the deviations obey another law of distribution. 
Therefore the inferences from Cornu's theorem are not binding. 

Finally, one point, often overlooked in textbooks and in scien- 
tific papers, must be stressed. A normal distribution is possible 
only if there is as stated earlier no effective reason that the 
deviations should not take all values from <*> to + This 
requirement is more or less fulfilled in the case of pressure, of air 
temperature, and of some derived elements. 

Many elements have zero as the limit on one side; e.g. precipi- 
tation, wind velocity, etc. In spite of this fact a more or less 
normal frequency distribution and inferences from it are not 
excluded if the frequency curve shows only relatively small 
frequencies in the vicinity of the given limit (e.g., zero). 9 

9 Naturally also skewness and kurtosis (peakedness, excess) have to be considered 
besides dispersion if an exact statement is to be made as to whether or not there is a 
Gaussian distribution. See the rain example in R. A. Fisher, Statistical Methods, p. 54 ff. 



42 



METHODS IN CLIMATOLOGY 



The precipitation during March in Helwan, Egypt (Table 8), 
offers an example of an extreme case. The deviations yield : 

a = 8.03 mm 

The arithmetic mean, however, is 6.3 mm of rain in March on the 
average for 21 years. 

TABLE 8. RAIN IN MARCH AT HELWAN, EGYPT 
(29.9N, 3i.3E, 38 FT.) 



Year 


A 

Amounts (mm) 


B 

Deviations 


Squares 6f the deviations* 


1904 





-6.3 


39.7 


5 


1.9 


-4.4 


19.4 


6 


4.8 


-1.5 


2.2 


7 


9.1 


2.8 


7.8 


8 


24.8 


18.5 


342.2 


9 





-6.3 


39.7 


10 


7.8 


1.5 


2.2 


11 


0.4 


5.9 


34.8 


12 





-6.3 


39.7 


13 


1.6 


-4.7 


22.1 


14 





-6.3 


39.7 


15 





-6.3 


39.7 


16 


25.8 


19.5 


380.2 


17 





> -6.3 


39.7 


18 


13.0 


6.7 


44.9 


19 





-6.3 


39.7 


20 


7.2 


0.9 


0.8 


21 


9.1 


2.8 


7.8 


22 


7.5 


1.2 


1.4 


23 





-6.3 


39.7 


24 


19.4 


13.1 


171.6 










2 




133.9 


1355.0 


rh 


6.3 















C. Deviations Grouped into Class Intervals 



Class interval 


Midpoint, mm 


Frequency % 


-7.0 to 


1.1 mm 


-4.0 


57 


-1.0 to 


4.9 


2.0 


24 


5.0 to 


10.9 


8.0 


5 


11.0 to 


16.9 


14.0 


5 


17.0 to 


22.9 


20.0 


9 



* Rounded off to the first decimal place. 

The negative deviations are crowded within the first class 
interval (Table 8, C) and the frequency curve has nothing to do 
with the normal distribution as seen in Figure 7. These conditions 
are exaggerated even more if the variate is bilaterally limited. 



SPECIAL STATISTICAL CHARACTERISTICS 



43 




-4.0 2.0 8.0 14.0 20.0 

mm 

FIG. 7. Frequency curve of rainfall in March (21 years) at Helwan, 
Egypt. On the X-axis: Midpoints of the class intervals of the deviations 
(mm) 




FIG. 8. Frequency distribution (per cent) of the degrees of cloudiness in ten 
September months at 9 p.m., at Prague, Czechoslovakia 



44 METHODS IN CLIMATOLOGY 

Perhaps the best example is the element cloudiness with its two 
limits : zero for cloudless sky, and ten for overcast sky. Figure 8 
shows the frequency distribution of cloudiness (in per cent) for 
ten months of September in Prague (50.1N, 14.4E, 663 ft.), 
at 9 P.M. The extremes are by far the most frequent values in 
contradistinction to the normal distribution, because of the 
bilateral limitation of the variate and of regional climatic proper- 
ties. 10 The arithmetic mean lies at about cloudiness 5, The 
arithmetic mean here loses its regular sense and indicates more or 
less the degree of mixing of the extreme values. 

To go back for a moment to the example of the March rain in 
Helwan: Cornu's theorem can be proved by means of the data 
given in Table 8. We get: 



or a value only 1% greater than Ludolfs number. This is a 
typical example of a series of an element with a frequency distribu- 
tion different on principle from the normal distribution. Never- 
theless, Cornu's theorem is nearly fulfilled. We repeat that it is a 
necessary and not a sufficient condftipn. 

III. 7. HIGHER CHARACTERISTICS (SKEWNESS) 

The higher characteristics are used to analyze frequency dis- 
tributions which are not identical with the normal distribution. 
For climatological purposes the asymmetry of the frequency curves 
is of interest. The term for it is skewness. In contradistinction 
to the symmetrical normal distribution, here arithmetic mean and 
mode no longer coincide. The arithmetic mean is much affected 
by extreme values of the variate ; the mode is not affected at all by 
the extremes. Therefore mean and mode cannot coincide. The 
greater the distance is between these two characteristics, the greater 
is the asymmetry, the skewness (Sk). According to what has 
been said before, the mean moves toward that side which shows 
the greater and more frequent extreme values. Therefore a 
frequency curve with the mean at the right side of the mode is 
called positively skewed (showing right skewness), while in the 
opposite case, it is negatively skewed (showing left skewness). 
Right and left skewness may be seen in Figures 9a and 9b. 

10 The statistics are taken from L. W. Pollak, in Prager Geophysikalische Studien, 
vol. VI (Prague, 1931), p. 11. 



SPECIAL STATISTICAL CHARACTERISTICS 



45 



K. Pearson gives as a measure of skewness the difference be- 
tween mean and mode. This difference has to be reduced to equal 
dispersion. Thus the equation results: 



Sk = 



Mean Mode 




MODE MEAN 



FIG. 9a. A right (positively) skewed frequency distribution showing the 
theoretical position of mode and mean. (After Arkin and Colton) 




MEAN MODE 
FIG. 9b. The same as above, but left (negatively) skewed 



It is useful to know, for instance, how far temperature devia- 
tions of one sign predominate as to quantity and to frequency. 
The trouble is that the position of the mode is rarely calculated, 
because of the great labor involved for long series. 



46 METHODS IN CLIMATOLOGY 

Less burdensome is Koppen's method of calculating the asym- 
metry (A). The equation reads: 



n 

where rib means the number of items below the mean, and n the 
total number of items. For the case of symmetry, A ~ since 
#6 = n/2. 

Example: C. F. Brooks gives the annual amounts of precipita- 
tion n for n = 80 years, at Philadelphia, Pa., expressed in devia- 
tions. The mean is added. 

A simple counting yields in a few minutes 

n b = 43 
therefore 

A - 1 - - 

A " l 80 " 

Compared with other results, the negative asymmetry is rather 
high. Negative deviations are a little more frequent. The ex- 
pectation of a relatively dry yearns somewhat greater than that 
of a wet year. 

The great advantage of Koppen's method is that it can be com- 
puted easily, and the A from different places are comparable with 
one another. On the other hand, it should not be forgotten that 
Koppen's method considers only the frequency of the signs, and 
that the amounts of the deviations are ignored. This is the great 
disadvantage of this method compared with the skewness method. 

Other characteristics of the frequency distribution, such as 
the peakness of the curve of frequency distribution, moments, 
etc., are of less interest to climatologists, and are not discussed 
here. 12 

III. 8. DIFFERENT KINDS OF VARIABILITY USED IN CLIMATOLOGY 

1) Mention was made earlier of average variability, which gives 
a good measure of the variations to which the climatological ele- 
ment is subject. 

U R. De Courcy Ward and C. F. Brooks, "The Climates of North America," in 
Kflppen-Geiger Handbuch der Klimatologie, vol. II, pt, J, p. 283. 

12 See R. A. Fisher, Statistical Methods; H. Arkin and R. R. Colton, An Outline of 
Statistical Methods (4th ed., New York, 1939); H. L. Rietz, Handbook of Mathematical 
Statistics (New York. 1924). 



SPECIAL STATISTICAL CHARACTERISTICS 47 

The average temperature of January has an average variability 

?\di\/n: 

0.7 F in Batavia, Java (6.2S) 

9.7 F in Green Harbor, Spitsbergen (78.0N) 

The average variability of the mean temperature of the months, 
is indeed climatically very characteristic. In the Arctic climate, 
the average variability of the temperature in January is about 14 
times greater than that in the inner tropic belt. 

2) If the series of observations is rather short (<10 years) the 
absolute range of variation of the series gives a first approximation 
of the variability of the element. 

3) The average variability is independent of the sequence of 
items. This is, however, of special climatological interest. It is 
important in judging of a climate, for instance, whether the changes 
from one day to another are rather smooth or very abrupt. 

This method is not restricted to equal sections of time (e.g., 
the day). It can be applied also to equidistant places along a 
straight line, running, for instance, from the coast inland. Special 
variabilities, regarding the sequence of the places in a given direc- 
tion, could yield new characteristics of a special contrast in the 
climate of a region. 

Thus a measure of variability which considers the quantity as 
well as the sequence of the items of the series of numbers character- 
izing the variations of a climatic element is necessary. 

Table 9 offers an example of the "average variability f> (AV) 
computed from the series in part A: A V = 5.0. 

In part B another variability (SV) is calculated in the following 
way: 

SV = [|0i - 8 1 + 1 02 - as | + + kn-i - 0|]r (n - 1) 

if n equals the number of items. This variability may be called 
inter-sequential variability, because its value is determined by the 
quantity of the items as well as by their sequence. In the ex- 
ample, SV = 8.8. 

For the sake of clarity, the series is arranged as an array in 
part C of Table 9, to show how great the influence of the order of 
numbers is upon the quantity of SV. This is now a quarter of 
that in part B. 

This important method of calculating a variability which takes 
into account the sequence of the items in the climatological series 



48 METHODS IN CLIMATOLOGY 

TABLE 9. COMPARISON BETWEEN AVERAGE VARIABILITY (AV) AND 
"INTER-SEQUENTIAL" VARIABILITY (SV) 

n * number of items, d = deviations from arithmetic mean (m). 
D =5 Difference between two consecutive items 

A 

Original order a\ oj a\ a\ a* a* &t a* a* aio 

of 10 numbers 9, 8, 17, 2, 9, 20, 0, 7, 12, 16; rh - 10 

deviations -1, -2, +7, -8, -1, +10, -10, -3, +2, +6 

S(J+) = 25, 2(d_) 25 



B 

Computation of SF 1, -9, 15, -7, -11, 20, -7, -5, -4 
from the series A 

Z(Z>+) - 36, S(Z>_) = - 43 



Q 

n 1 9 



C 

Series of A arranged 0, 2, 7, 8, 9, 9, 12, 16, 17, 20 

as array 
Differences between 2, -5, 1, -1, 0, -3, -4, 1, -3 

consecutive items 

>+) . 0, *(ZU - - 20 






can be applied to each element as far as it represents a scalar 
quantity completely specified by one number, in contradistinction to 
directional quantities, which have direction as well as magnitude 
(wind, motion of clouds, motion of the water in an ocean cur- 
rent, etc.). 

The best-known example of an inter-sequential variability is 
the interdiurnal variability. This is calculated by two methods: 

1) Differences of the consecutive daily means of the element; 

2) Differences of the element at a fixed hour of consecutive days. 

The advantage of the first method is that, for instance, the 
mean daily temperature is derived from at least two observations 
(e.g., the extremes), so that incidental errors are not so likely. 
On the other hand, average values can always conceal interesting 
phenomena. 

The disadvantage of the second method is that, for instance, 
the interdiurnal variability of temperature has a well-accentuated 



SPECIAL STATISTICAL CHARACTERISTICS 49 

daily variation. The greatest trouble is that the time of the ex- 
tremes is variable in the course of the year. 

In Central Europe (Potsdam), the maximum interdiurnal 
variability occurs at 6 A.M. in October, and at 4 P.M. in April. 
This daily variation reaches its maximum in July, when the ratio 
maximum/minimum is 1.8. Much caution should be exercised 
before drawing conclusions from investigations in interdiurnal 
variability taken from a fixed hour. 

Tables 10 gives a nearly complete example of an investigation 
in interdiurnal variability of temperature. 13 



TABLE 10. DIFFERENT STATISTICS REGARDING INTERDIURNAL 
VARIABILITY OF TEMPERATURE 

Col. 2 = Interdiurnal Variability of Temperature (5 A.M.) (ID V) at Mount Washington, 

N. H. (6270 feet). 
Col. 3 = Average Coolings (C) 
Col. 4 = Average Warmings (W) 
Cols. 5 and 6 = Average Numbers of Consecutive Days with Increasing (WD) and 

Decreasing (CD) Temperature 
Col. 7 = Length of Temperature Surges in Days (S) 

(After V. Conrad) 



0) 

Month 


(2) 
IDV 
F 


(3) 
C 
F 


(4) 
W 
F 


(5) 
CD 
days 


(6) 
WD 
days 


(7) 

5 
days 


Dec 


8.7 


9.2 


8.3 


1.84 


1.92 


3.76 


Jan 


11.4 


13.8 


9.6 


1.44 


1.86 


3.30 


Feb 


10.1 


9.9 


10.4 


2.36 


2.45 


4.81 


Mar 


. . ..98 


104 


9 2 


1.79 


1.92 


3.71 


Apr. 


62 


66 


5.9 


1 62 


2.13 


3.75 


May 


5 9 


63 


5 6 


1 79 


2.13 


3.92 


June 


5.1 


5.2 


5.0 


1.79 


1.95 


3.74 


July 


4.0 


4.4 


3.7 


1.74 


2.13 


3.87 


Aug 


4.2 


4.4 


4.1 


2.05 


2.40 


4.45 


Sept 


6.3 


7.5 


5.2 


1.87 


2.13 


4.00 


Oct 


8.0 


8.3 


7.7 


2.09 


2.23 


4.32 


Nov 


8.0 


8.5 


7.5 


1.96 


2.04 


4.00 



(rft) 7.3 7.9 6.8 1.86 2.11 3.97 



Column 2 contains the interdiurnal variability (IDV) at 5 A.M. 
The values are averages of 3 years. Irregularities of the annual 
course of ID V, as well as of the elements in the other columns, are 
to be explained by the shortness of the period under discussion. 

18 V. Conrad, "The Interdiurnal Variability of Temperature on Mount Washing- 
ton," Transactions of the American Geophysical Union (Washington, 1942), pt. II, p.279. 



50 METHODS IN CLIMATOLOGY 

The statistics which lead to the IDV (which ignores the signs) 
enable one to calculate separately the average amount of tempera- 
ture increasing (W) (col. 4), or decreasing (C) (col. 3) from day 
to day. The increase of temperature from one day to the next 
might be termed warming; the decrease, cooling. 

Because in the problem in question the amounts of cooling 
and of warming are not effectively influenced by radiation proc- 
esses, they are a measure, or at least an indication, of advected 
cold and warm air masses. Therefore, these statistics are valuable 
for climatology as well as for long-range forecasting, etc. When 
the IDV is calculated, average and maximum values of cooling and 
warming should be published. This can be done with little or no 
supplementary labor. 

A further valuable addition is represented by computing the 
number of consecutive days with increasing temperature (WD) 
(column 6), and with decreasing temperature (CD) (column 5). 
The individual numbers are averaged for each month. 

Column 7 shows the sums of CD + WD : that is, the average 
time in days for an average, consecutive increase and decrease. 
This seesaw can be called a temperature surge, and column 7 (S) 

gives the average duration of the surges in days. 

% 

Once more, it should be emphasized that the inter-sequential 
variability can and should be used when different climatological 
elements are discussed. For lack of space, only one example of 
temperature and the time-unit of one day is given in this chapter. 

For one purpose or another, an interhourly variability may be 
useful. As an example, the wind velocity (regardless of the direc- 
tion) could be studied from this standpoint, if the establishment of 
windmills in a given locality were planned. 14 B. M. Varney 15 uses 
an interannual variability to characterize the rain conditions of 
California. Annual and seasonal amounts are both handled by 
this method. It goes without saying that monthly average data 
for air pressure, temperature, cloudiness, vapor pressure, precipita- 
tion, depth of snow, etc., can be studied also by interannual, inter- 
seasonal, etc., variability. 

14 See also V. Conrad, " Interhourly Variability of Temperature at Mount Washing- 
ton, N. H.," Transactions of the American Geophysical Union, 1943, p. 122. 

15 B. M. Varney, "Seasonal Precipitation in California and its Variability," Monthly 
Weather Review, 1925, pp. 148-163, and 208-218. 



SPECIAL STATISTICAL CHARACTERISTICS 51 

III. 9. ABSOLUTE AND RELATIVE VARIABILITY AND OTHER 
MEASURES OF VARIATIONS 

If amounts of such as precipitation are considered, the average 

variability ^ - - = AV depends naturally upon the arithmetic 
n 



mean 

The normal annual precipitation in Cairo, Egypt, is 1.34 in.; 
the variability is 0.67 in. This is large, because the annual pre- 
cipitation varies between one half and three halves of the mean. 
The same variation in Rangoon, Burma (17N,96E), with a 
4 4 normal " precipitation of about 100 in., would be negligible indeed. 

Therefore, a new concept is necessary, no longer dependent on 
the arithmetic mean : the relative variability. This is the absolute 
variability expressed in per cent of the arithmetic mean. 



pup 

The relative variability V r would reduce the absolute variability 
AV to the unit of annual precipitation only if the correlation be- 
tween the two variants, V r and p, were linear. This assumption 
does not hold, as is shown from the observations. 

Statistics from about 360 stations scattered over the earth 
yielded a correlation between V r and normal rainfall. This is 
represented in Figure 10 by a hyperbolic curve. Therefore, the 
influence of the normal sum of precipitation becomes weak only 
beyond the turning-point of the curve, i.e., if the normal rainfall is 
larger than about 28 inches. Then, in a first rough approximation, 
V r is practically no longer dependent upon p. It should be stressed 
that about 40% of the earth's surface has less than 28 inches pre- 
cipitation, and 30% has less than 20 inches. Conclusions drawn 
from comparing values of V r for different places in these vast 
regions of small annual precipitation are inaccurate and mislead- 
ing. 17 A way out of these difficulties is offered by the representa- 

16 If the arithmetic mean is calculated from a longer period, perhaps at least 25 
years, it is called normal value or normal. The Meteorological Office in London pub- 
lished a Book of Normals based on a period of 35 years (see Xll.S.f). 

17 For more details, see V. Conrad, "The Variability of Precipitation,' 1 MWR, 
vol. 69 (1941), pp. 5-11. Statistics taken from E. Biel, "Die Ver^nderlichkeit der 
Jahressumme des Niederschlags auf der Erde, M Geographisches Jahrbuch aus Oesterrcich, 
14/15, 1929. 



52 



METHODS IN CLIMATOLOGY 



tion of anomalies, treated later (XIV, 7). In regions with an 
annual precipitation greater than about 20 to 28 inches, the values 
of V r can be compared with one another without serious error. 



Vo 

90 
SO 

60 



20 




20 



40 



60 80 

INCHES 



FIG. 10. Relative variability and rainfall. (After V. Conrad)* 
* v f in the diagram should read V r . 

Since the calculation of V r is rather troublesome for long series 
of observations in many investigations of precipitation conditions, 
a simpler measure of the variability is commonly used. G. Hell- 
mann introduced an expression which is called ratio of variation 
(Q) - 18 It is nothing but the quotient 



m 



where M = the largest annual amount in the series and m = the 
smallest. This measure should not be used, since Q becomes 
infinite in many places located in the desert belts. 

18 G. Hellmann, Veroffentlichungen Preuss. Met. Inst. Ill (1909), no. 1. 



SPECIAL STATISTICAL CHARACTERISTICS 53 

E. Gherzi used another measure of variation : 19 

_ M m 
V p 

It is a great improvement over the measure M/m. 

III. 10. SOME APPLICATIONS OF THE 
METHOD OF RANDOM SAMPLES 

III. 10. a. Precipitation 

Though the following inferences are restricted to one or another 
element (precipitation and fog) similar methods could be applied 
to obtain results for other elements. 

At some thousands of climatological stations, eye observations 
are made from one to three times a day, but no continuous records 
are kept. One assumes that the observer enters the international 
sign for rain or snow in his register when it rains or snows at the 
precise time of the observation. Then, the observation can be 
valid for this time as a random sample of the actual weather. 
Starting from this idea, W. Koppen derived from these instantane- 
ous observations data which, otherwise, could be taken from con- 
tinuous records only. Probability has previously been defined as 
the ratio of the number of favorable or desired events to the total 
number of possible events. If one is interested in precipitation, 
observations, just at the moment of rain or snow, are the favorable 
events. In a given period (e.g., a month), n observations are 
made. Among these, r observations occur exactly during the 
precipitation, Then, the proper fraction r/n = p, is the absolute 
probability of precipitation; " absolute/' because independent of the 
selected unit of time. The absolute probability of precipitation is 
the basis of Koppen's estimates. 

The following statement and some simple inferences give a 
survey of the quantities in question and a demonstration of how 
to use them. 

n = Number of all observations within the chosen period 

(e.g., month, season) 
r = Number of all observations with precipitation 

19 p = average amount of precipitation of the period in question. 

80 E. Gherzi. tude sur la pluie en Chine (Observatoire Zi-Ka-Wei, 1928). 



54 METHODS IN CLIMATOLOGY 

N = Total number of hours in the chosen period (e.g., 720 

hours in month of June) 
d = Number of days with precipitation within the chosen 

period 
h = Amount of precipitation in the period in question 

y 

p = - = Absolute probability of precipitation 
n 

Then : 

D = pN = Probable total duration of precipitation in hours 

D pN , ... 

= j- = Average duration of precipitation, per day with 

precipitation, in hours 

h 

- = "intensity" of rain (average amount of rain per rainy 

d day) 

= h rr = Average amount of rain in one rainy hour 
D prJ 

Example: At the observatory in Batavia (Java), not only were 
all the necessary observations made by eye and by self-recording 
gauges, but also the data were published in full detail, so that the 
method described can be tested to see how far it is in agreement 
with observation. 

January 1915 21 was the month of the test (dry period). 
Regular observations were made three times a day: n = 93 
By counting, it was determined that: r = 10 
Therefore: p = r/n = 10/93 = 0.1075 

Further: N = 31 X 24 =744 hours 
Hence : D = pN = 80 hours with rain 

Observed: d = 29 

Therefore: D/d = 80/29 = 2.76 hours 

Observed: h = 13.07 inches 

Hence: h/d = 13.07/29 = 0.45 inches/rainy day 

and h/D = 13.07/80 = 0.163 inches/rainy hour 

From Table 11 it can be seen how well this method given by 
Koppen agrees with observation. The computation is simple and 
the method is easy to understand. On the other hand, it supplies 
information about data which as stated before can usually be 

21 Yearbook of Batavia, Java. 



SPECIAL STATISTICAL CHARACTERISTICS 55 

TABLE 11. METHOD OF RANDOM SAMPLES APPLIED TO RAINFALL AT 

BATAVIA, JAVA 

Number of Average duration of rain in Average amount of rain 

rainy hours one rainy day in one rainy hour 



D 

obs. 


D 
calc. 


D/d 
obs. hours 


D/d 
calc. hours 


h/D 
obs. in. 


h/D 
calc. in. 


75.3 


80 


2.6 


2.8 


0.173 


0.163 



obtained only from continuous records. These ideas could be used, 
with the necessary modifications, for other climatological elements. 

III. 10. b. Cloudiness (C) and Duration of Sunshine (S) 

Another problem using random samples may be described. 
At every climatological station, cloudiness is observed. At a few 
of them, bright sunshine is recorded continuously. It goes without 
saying that the average monthly duration of sunshine is most 
interesting information for many purposes. 




FIG. 11. The relation between cloudiness and duration of sunshine. 
E abcdef W = path of the sun. The parts of the path ab } cd, ef, are covered 
by clouds. 

The question of how to calculate sunshine duration from aver- 
age cloudiness is approximately solved by means of random sam- 
ples. We assume that the path of the sun (Fig. \\.Eabcdef W} 
for ajjiyen day of the year is obscured by clouds for the portions 
a6> cd> ef. If these parts are known from the records of a sunshine 
recorder, a simple calculation gives the percentage, C, of the total 



56 METHODS IN CLIMATOLOGY 

length of the path of the sun which is covered by clouds. It is 
clear that the rest of the path S = 100 - C. 

Now, if the assumption is made that the conditions along the 
path of the sun are a random sample, valid for the entire visible 
sky, the problem is solved in principle. In reality, this hypothesis 
cannot be made for an individual day, only for an average day 
within a period of at least one month. Even then, the assumption 
remains a rough approximation, as will be grasped from the ex- 
amples in Table 12. They show that the annual and monthly 
values of S and C yield sums which indeed generally differ less 
than 10% from 100. This agreement is not bad. 

The simple formula, S + C = 100%, has been improved by 
theoretical considerations. The resulting formula is not tested 

TABLE 12. THE EQUATION S + C = 100% COMPARED WITH OBSERVATION RESULTS 
IN CALIFORNIA AND IN THE MEDITERRANEAN REGION 

(S in % of possible duration; C in % of the visible sky) 

Jan. July Year 

% % % 

San Francisco, Calif. S: 53 69 65 

37.8N, 122.7W C: * 54 37 42 

S + C: 107 106 107 

San Diego, Calif. S: 67 68 68 

32.7N, 117.2W C: 40 34 36 

S + C: 107 102 104 



Sacramento, Calif. 
38.6N, 121.4W 


S: 
C: 


46 
59 


96 
08 


74 
31 



S + C: 105 104 105 



Messina, Italy S: 37 75 52 

38.2N, 15.6E C: 61 28 46 

5 + C: 98 103 98 

Athens, Greece S: 49 81 60 

38.0N, 23.7E C: 55 11 40 



5 + C: 104 92 100 



Helwan, Egypt S: 70 90 82 

29.9N, 31.1E C: 41 06 23 

S+C: 111 96 105 



SPECIAL STATISTICAL CHARACTERISTICS 57 

for different climates, and its evaluation is relatively complicated, 
so that, generally, the climatographer would prefer not to use it. 22 
Finally, in connection with this problem, another definition 
may be mentioned : 

B = 100 - C 

where C is again the cloudiness in per cent of the visible sky ; B can 
be called the brightness of the sky, and is used sometimes in meteor- 
ological and climatological papers. 

11 See V. Conrad, " Die klimatologischen Elemente und ihre Abhangigkeit von 
terrestrischen Einfltissen," Handbuch der Klimatologie, vol. I (Berlin, 1936), p. 450, and 
A. Wagner, "Beziehungen zwischen Sonnenschein und BewOlkung in Wien," Meteoro- 
logische Zeitschrift, 1937, pp. 161-167. 



CHAPTER IV 

SOME PROBLEMS OF CURVE FITTING AND SMOOTHING OF 

NUMERICAL SERIES 

IV. 1. THE STRAIGHT LINE 

IN CLIMATOLOGY, one is often confronted with the. problem of 
representing an element as a function of an independent 
variable, such as altitude, latitude, distance, etc. Each value 
of the variable element is related to a certain value of the inde- 
pendent variant. The first step is to arrange the values of the 
element according to increasing values of the independent variable. 
If there is a greater number of values, it may be advantageous to 
divide the data into groups, and to average either variant within 
each interval. 

The variables are plotted on the X- and F-axis, respectively, 
and a line is drawn freehand through the resulting points. 

This procedure may suffice for survey of the correlation of both 
variants. In any case, the type of* curve can be recognized; in 
climatology, one may often assume it to be a straight line. 

TABLE 13. THE RELATIONSHIP BETWEEN THE AMPLITUDE OF THE ANNUAL VARIATION 

OF AIR PRESSURE (ai, in mm mercury) AND THE ALTITUDE ABOVE SEA LEVEL 

(h in hectometers; 1 hectometer = 100 m) 

(1) (2) (3) (4) (5) (6) 

h a\ a\ a\ 

(calculated by 
the semi-average 

(mm Hg method Obs. calc. (calc. with 

No. (hm) observed) a\ =0.29 +0.16/0 (2-3) 01 =0.36 +0.16/0 Obs. -calc. 



1 


0.5 


0.28 


0.37 


-.09 


0.44 


-.16 


2 


2.7 


1.08 


0.72 


+ .36 


0.79 


+ .29 


3 


4.6 


1.04 


1.02 


+ .02 


1.09 


-.05 


4 


6.1 


1.24 


1.26 


-.02 


1.33 


-.09 


5 


7.8 


1.38 


1.54 


-.16 


1.61 


-.23 


6 


9.0 


1.72 


1.73 


-.01 


1.80 


-.08 


7 


16.4 


3.04 


2.91 


+ .13 


2.98 


+ .06 


8 


23.4 


4.26 


4.03 


+ .23 


4.10 


+ .16 


9 


30.5 


5.34 


5.17 


+ .17 


5.24 


+ .10 


2(+) 








+91 




+ 61 


s(-) 








-28 




-61 



+63 

58 



SOME PROBLEMS OF CURVE FITTING 



59 



An example illustrates this problem : Records of the whole-year 
amplitudes of air pressure at 3 1 stations are correlated with the alti- 
tude of the respective places. The amplitudes are arranged in 9 
groups according to increasing heights. The averages (hectom- 
eters) of the intervals are indicated in column 1 of Table 13. The 
averages of the respective amplitudes are contained in column 2. 



30 
28 
28 
24 
22 

20 

18 



I- 



ie 
to 

8 
8 

4 
t 




z 



1.0 2.0 3.0 4.0 

AMPLITUDE O.I mmHg 



8.0 



FIG. 12. Annual range of air pressure (abscissa) and altitude (ordinate). 

(After V. Conrad) 



Starting from these two series of numbers, one derives the 
points marked on the graph (Fig. 12). A straight line fits the ob- 
servations best. 

The analytical equation of a straight line has the form : 

ai = A + Bh 

where a\ means the amplitude of the whole annual variation, h the 
altitude, and A, B, constants which have to be evaluated. For 
this reason 9 equations are available from Table 13. 



60 METHODS IN CLIMATOLOGY 

No. 1 0.28 = A + 0.5 B 
No. 2 1.08 = A + 2.7B 

No. 9 5.34 = A + 30.5 B 

Here is just the case to use the method of least squares. 

As was said above, however, as a rule it does not pay to apply 
this rather wearisome method to climatological problems. Usu- 
ally, the accuracy of the observations is not sufficient to use least 
squares. 1 Therefore a method of quick approximation is given 

1 In the case of a linear relation, the method of least squares yields relatively simple 
equations for the constants A and B of the equation a = A + Bh, the computation of 
which is not too laborious. 

Sfr-S(qft) - 2a-2fr* _ Sft-2a - nZ(ah) 



These formulas, applied to the example of Table 13, yield: (1) for the calculation 
of the numerator of A : 

Xh = 101.0; 2(aA) = 354.158; 
2h-X(ah) = 35769.958; 
Sa = 19.38; S 2 = 1954.52 
2a-2^ = 37878.5976 

(2) for the calculation of the numerator of B: t 

Sfc-Sa = 1957.38; w-Sfa/i) * 3187.422 

(3) for the denominator of A and B : 

(Sfc) 2 = 10201.0; n-2fc' = 17590.68 
and finally: 

A = + 0.2853 
B = + 0.1664 

so that the equation, got by the method of least squares, reads: 

ai = 0.285 + 0.166& (I) 

The semi-average method, given in the text, resulted in: 

0! 0.29 + 0.16A (II) 

The values of A are nearly identical, the values of B differ for about 4%. For further 
correction, the straight line defined by equation (II) is shifted parallel to itself increasing 
A from 0.29 to 0.36 and the equation reads 

ai 0.36 + 0.16A (III) 

This step is advantageous in so far as it makes the sum of the positive deviations equal 
to that of the negative ones by a minimized arithmetic. In the present example it would 
have been more nearly correct to leave the constant A unchanged and to diminish the 
angle between the straight line and the abscissa as it can be seen from the exact equation 
(I). On the whole, the agreement between the approximations given by the semi- 
average method and similar methods, on the one hand, and the exact result of the method 
of least squares, on the other hand, is generally sufficient for climatological purposes. 
The reader who takes pains with making a diagram on graph paper to a scale even twice 
that of Figure 12 will become aware of how close the three lines run to each other. 



SOME PROBLEMS OF CURVE FITTING 61 

here instead, the semi-average method. The first half and the second 
half of the values of both variables are averaged, so that two pairs 
of coordinates result, which have to be introduced into the general 
equation of a straight line. 

In the special case of Table 13, the number of items is relatively 
small and odd. In order to avoid asymmetry and unequal weight, 
item no. 5 can be included both in the first and in the second half. 

The first half then contains items nos, 1 to 5 ; and the second 
half, nos. 5 to 9. 

The two equations read : 

1.004 = A + 4.345 
3.148 =4 + 17.42 B 

Therefore A = 0.29 and B = 0.16 and the straight line is given 
byai = 0.29 + 0.16 A. 

The nine values of h (Table 13, nos. 1 to 9) introduced into the 
equation yield the calculated values of #1 in column 3. Column 4 
contains the differences: Observed values (obs.) minus calculated 
values (calc.). It is seen that (at the bottom of the table) the 
positive differences exceed the negative ones by 0.63. This means, 
that the observed a\ are, on the average, 0.07 mm Hg greater than 
the calculated. Therefore, we have to shift the calculated straight 
line parallel to itself to the right side by 0.07 mm: in other words, 
increase the constant term 0.29 by 0.07. 

The definitive equation now reads: 

ai = 0.36 + 0.16 h 

The new values computed with this equation appear in column 
5, and the differences "obs. calc." in column 6. Incidentally, 
the sum of the differences (with regard to the signs) is zero, which 
represents the ideal. The process of shifting has to be continued 
until the sum becomes a minimum. 2 In many cases it is not even 
necessary to use the semi-average method. 

When the number of observations varies greatly from one 
class interval to the next, it is sometimes more advantageous to 
take the averages of only two intervals with the greatest number of 

2 The example is taken from V. Conrad, "The Influence of Altitude on the Yearly 
Course of Air Pressure," Bulletin of the American Meteorological Society, vol. 20 (1939), 
p. 207. 



62 



METHODS IN CLIMATOLOGY 



observations, in such a way that the midpoints of the intervals may 
be as far apart as possible. 3 

IV. 2. DECAY-CURVES 

In climatology, as in some other fields, the observation of the 
decay of any quantity intensity, ratio, etc., with time, distance, 
height, or depth, has to be represented by an analytical equation. 

Example: In a problem dealing with cloudiness 4 the decay of a 
ratio y is to be described which is infinite at the place of observa- 
tion and decays to zero with increasing distance. These boundary 
conditions lead to the equation 

y = Ax~ B 
In terms of logarithms: 

log y = log A B log x 

Thus, the exponential equation is reduced to a linear equation 
regarding the two unknown constants log A and B. Plotting on 
logarithmic paper yields the values A and B in more or less rough 



2.00 



1.80 



1,60 



140 




1.20 



30 40 60 80 100 
DISTANCE km.x 



120 



FIG. 13. 



Observations represented by the formula y 
(After V. Conrad) 



Ax 



-B 



* For an interesting case of curve fitting, see Hans Neuberger, "Studies in Atmos- 
pheric Turbidity," Pennsylvania State College Studies, no. 9 (1940), p. 22. There, a dot 
chart is presented which seemingly offers a complex problem. In reality, it is easily 
solved by the assumption of two straight lines instead of one curve of higher order. 

'Taken from V. Conrad, "Zum Studium der Bewolkung," Met. Z.S., 1927, p. 87. 



SOME PROBLEMS OF CURVE FITTING 63 

approximation. On the other hand, the problem is reduced to that 
of the straight line. This has been discussed above. Therefore 
the numerical example in Table 14 and in Figure 13 does not need 
further explanation. 

TABLE 14. THE REPRESENTATION OF OBSERVATIONS BY AN EXPONENTIAL CURVE 

y = Ax~ B 



(1) 


(2) 


(3) 


(4) 


Distance * 
(km) 


Ratio y 
obs. 


Ratio y 
calc. 


Obs. - calc. 


21 


2.09 


2.10 


-.01 


48 


1.65 


1.69 


-.05 


80 


1.60 


1.48 


+ .12 


123 


1.27 


1.33 


-.06 


2(+) 


+ .12 


s(-) 






-.12 



The solution of the logarithmic equation yields : 

A = 4.618 B = 0.2590 
and the equation reads : 

y = 4.61 8 x-o.2590 

IV. 3. THE EQUATION y = B-e Ax 

If the dependent variable, y, has a finite value at the origin and 
increases with increasing x to infinite values, the equation 

y = B-e Ax 

may often represent the observations in a suitable way. Plotting 
on semilogarithmic paper yields the values of A and B. 
In logarithmic terms the equation reads: 

log y = log B + A x log e 

where "log" means common (Briggsian) logarithms, e = 2.718, 
log e = 0.4343. The logarithmic equation has a linear form so 
that A and B can be easily determined. 

No general rule can be given for the choice of the right form of 
equation. The graphical representation remains the best guide. 
A collection of different forms of equations for curve fitting can be 



64 



METHODS IN CLIMATOLOGY 



found in various books dealing with numerical calculations and 
mathematical statistics. 6 

IV. 4. THE QUADRATIC EQUATION 

Figure 14 represents an example of a group of curves which is 
also important in the discussion of climatic problems. The ob- 
servations averaged for 5 class intervals show the behavior of 

HM 








FlG. 14. Observations represented by a quadratic equation 
(x = A + By + Cy*) 

(After V. Conrad) 

E.g., in H. L. Rietz, Handbook of Mathematical Statistics (New York, 1924), 
where also the transformation of the different equations to linear forms is given. 

Different forms of the equation for the hyperbola (e.g. xy = const, or y 2 = x(m + nx) 
are often very suitable for the representation of the correlation between climatological 
elements and factors. Examples are given in V. Conrad, " Messung und Berechnung 
der AbkiihlungsgrGsse" (Measuring and Calculating the Cooling Power; with an English 
summary), Gerlands Beitrage zur Geophysik, vol. XXI, 1929, pp. 183-189. 

Another example of curve fitting by means of hyperbola equations can be found in 
V. Conrad, and O. Kubitschek, " Die Veranderlichkeit und Machtigkeit der Schneedecke 



SOME PROBLEMS OF CURVE FITTING 65 

average winter temperature at different average heights above the 
bottom of an Alpine valley. 6 The well-known phenomenon of the 
inversion is illustrated by means of the numbers in Table 15 and 
by Figure 14. 

TABLE 15. OBSERVATIONS REPRESENTED BY THE QUADRATIC EQUATION: 

x = A + By + Cy (VARIATION OF TEMPERATURE WITH 

HEIGHT IN AN ALPINE VALLEY IN WINTER) 

(h = hectometers, / in C) 



(1) 


(2) 


(3) 


(4) 


h 


fC 


/c 




No. (hm) 


(observed) 


(calculated with 
/=-8.1-fl.06A-0.05A) 


Obs. -calc. 


1 4.9 


-4.0 


-4.1 


+0.1 


2 6.7 


-3.1 


-3.2 


+0.1 


3 9.0 


-2.5 


-2.6 


+0.1 


4 11.3 


-2.9 


-2.5 


-0.4 


5 19.0 


-5.8 


-5.9 


+0.1 


2(+) 






+0.4 


2(~) 






-0.4 



In cases like this, it is not easy to determine the exact nature of 
the curved line; so the use of a quadratic equation is recommended. 
The form is: 

x = A + By + Cy 2 

where y is the independent variable. 

According to Table 15, five pairs of values can be used for com- 
puting the coefficients A, B, C. 

Even in this problem, the use of the method of least squares 
can be avoided if the observed values lie in a smoothed curve, as 
is seen in the actual example. Then a good approximation can 
often be reached by choosing only three equations for computing 
the three constants. Naturally, points on the curve which charac- 

in verschiedenen Seehohen" (Variability and Depth of Snow on the Ground in Different 
Altitudes, with an English summary), Gerlands Beitrdge zur Geophysik, vol. LI, 1937, 
pp. 100-128. In the East Alps, the variation of the depth of snow on the ground with 
altitude can be well represented by hyperbolas from November through March. In 
April, and this is of great methodological interest, the hyperbola must be replaced by a 
parabola, which considers the melting of the snow cover in the lower levels in spring, 
on the one hand, and a further increase of the depths of snow in the high levels, on the 
other hand. The hyperbola reproduces the relations between depth of snow and alti- 
tude as far as in every level an increase of the depths takes place. 

8 Example from V. Conrad, Klimatographie von Karnten (Wien, 1913). 



66 METHODS IN CLIMATOLOGY 

terize the trend as well as possible should be selected from the 
graph. 

In the present example, the following equations were chosen: 

-4.0 = A + 4.9 B + 24 C 
-2.5 = A + 9.0 B + SIC 
-5.8 = A + 19.0 B + 361 C 

These are the pairs of values nos. 1,3,5. The equations are easily 
solved. It is necessary to change the term A by only 0.1 C to 
equalize the sum of positive and negative differences "obs. calc." 
It follows the equation : 

/ = - 8.0 + 1.05 h - 0.05 A 2 (C) 

where t means the temperature, h the height in hectometers. The 
constants are : 

A = - 8.0; B = 1.05; C = - 0.05 

The differences "obs. calc." show clearly that, in general, in 
a climatological problem greater accuracy is unnecessary ; another 
choice of intervals, another choice of stations, etc., would cause 
slight changes of the vertical temperature distribution which are 
of the same order of magnitude as the differences "obs. calc." 
Therefore the application of more exact methods would be a waste 
of time. 

The advantages of representing variations of elements by 
analytical equations are obvious. The equations permit an objec- 
tive interpolation, based on the totality of observations. 

IV. 4. a. Level of the Turning Point of the Curve 

As a supplement to the last problems, two applications of the 
representation of observations by means of analytical equations 
are given. From the observations (see Fig. 14), it is clear that the 
temperature increases in the lower layers with height (inversion), 
and above a certain level the normal decrease of temperature 
occurs. 

Question : At what altitude is the turning point of the tempera- 
ture-height curve located? 



SOME PROBLEMS OF CURVE FITTING 67 

At the extremes, the differential quotient of a function becomes 
zero. 

^ = 1.05 - 2 X 0.05 h = 1.05 - 0.1 h 
dh 

3/J = 1.05 - 0.1 A = 

dfl / t~max 

and hence (/*)<-ma* = 10.5 hectometers, or = 3445 ft. 

This level is the average top-level of the inversion in winter in the 

mountain valley in question. 

IV. 4. b. The Standard Distribution 

As was said before, the analytical equation permits full inter- 
polation of the element for every value of the independent variable. 
In the present example, the average winter temperature can be 
given for every desired altitude which lies between the extreme 
levels of the places of which temperatures are used in the com- 
putation. 

For some purposes, it is very important to know the average 
temperatures at the consecutive levels, for instance, 10-meter or 
100-meter levels, above the bottom of the valley. 

A clipping from such a table calculated from the equation 

t = - 8.0 + 1.05 A -0.05/t 2 
is given in Table 16. It indicates the average temperatures every 

TABLE 16. CLIPPING FROM A TABLE OF AVERAGE DISTRIBUTION OF TEMPERATURE 
ABOVE THE BOTTOM OF A MOUNTAIN VALLEY IN JANUARY 



Altitude 


00 




10 




20 




30 




40 etc. 


meters 










Centigrade 


degrees 








400 


-4 


.6 


-4, 


5 


-4. 


5 


-4 


.4 


-4.4 


500 


-4 


.0 


-3.9 


-3. 


9 


-3 


.8 


3 ft 

"^ O.O 


600 


-3 


.5 


-3. 


5 


-3. 


4 


-3 


.4 


-3.3 


etc. 





















10 m. The content of such a table represents the standard distribu- 
tion of temperature with height, in the region in question. 

This representation of the distribution of temperature with 
height is naturally only one example out of many. The method 
can and should be applied to any climatological element which is 



68 METHODS IN CLIMATOLOGY 

correlated with a climatic factor, or with another climatic element. 
(See XIV, 4 and 5.) 

IV. 5. SMOOTHING OF NUMERICAL SERIES 

This procedure is usual in climatological practice, if the aver- 
aged variation of an element for instance, an average annual 
course presents an irregular aspect. One assumes that the ir- 
regularities are of incidental nature and would diminish in proportion 
to the length of the available period. 

Different methods are applicable. Some, of greater importance 
for the climatologist, are mentioned below, in each of which a 
series of the following ten numbers will be used for a numerical 
example: 12, 31, 9, 4, 11, 10, 15, 7, 11, 13. 

1) Overlapping sums: 

(a) of three consecutive elements according to the formula : 

di = (#;_! + 0; + 0;+i) 3 

where means any element in the series, the subscript (i 1) 
stands for the preceding, (i + 1) for the following, element. 
di is the smoothed value replacing thfe^original a*. Example: 

01, 02, 03, 04, 05, 06, 07, 08, 09, 010 

12, 31, 9, 4, 11, 10, 15, 7, 11, 13 

The series may be considered as cyclic (for instance, an annual 
course). Then, the smoothing procedure begins with the last 
item and ends with the first, and the smoothed series reads: 

01 = (010 + 0i + 02): 3 = (13 + 12 + 31): 3 = 19 

2 = (01 + 02 + 03): 3 = (12 + 31 + 9): 3 - 17 

010 = (09 + 010 + 01): 3 = (11 + 13 + 12): 3 = 12 
so, finally, we have the sequence of smoothed values : 
19, 17, 15, 8, 8, 12, 11, 11, 10, 12 

(b) The number of overlapping consecutive items can be in- 
creased arbitrarily, considering the total number of items of the 
series. For simple reasons, odd numbers should be preferred. As 
a second example, the formula for 5 consecutive items is given: 

0; = (0i-2 + fli-i + + 0;+i + 0*4-2) 



SOME PROBLEMS OF CURVE FITTING 69 

The item di of the series (la) is therefore: 

a\ = (ag + aio + #1 + #2 + #3) : 5 
= (11 + 13 + 12 + 31 +9): 5 = 15 

a 2 = (#10 + fli + #2 + #3 + #4) : 5 

= (13 + 12 + 31 + 9 + 4) : 5 = 14 etc. 

2) Weighted overlapping sums: 

(a) <z* = (a;_i + 2#i + a t -+i) : 4 
Numerically : 

#1 = (^10 + 2&i + a 2 ) : 4 

= (13 +2 X 12 + 31): 4 = 17 etc. 

(b) a* = (a t -_ 4 + 2a l - 3 + 3a v _ 2 + 4ai_i + Sa t - +1 . 25 

ai+4 + 2a i4 . 3 + 3a i+2 + 4a i+ i) j ' 

Numerically : 

di = (a 7 + 2a 8 + 3a 9 + 4ai + ^ a i + 1 . 25 
a 5 + 2a 4 + 3a 3 + 4a 2 ) j ' 

fii = (15 + 14 + 33 + 52 + 60 + 

11 + 8 + 27 + 124) = 14 etc. 

(c) A more individual weighting is obtained by using the bino- 
mial coefficients as weights for the consecutive items. Formula 2a 
represents the simplest case, the binomial coefficients of a squared 
binom : 

i.e., 1, 2, 1 

The binomial coefficients for the 4th power are: 1, 4, 6, 4, 1; and 
those for the 6th power: 1, 6, 15, 20, 15, 6, 1. 

Therefore, the smoothing formula, corresponding to the 4th 
power, is: 

di == (ai_ 2 + 4a i _ 1 + 6a.- + 4a i+1 + a i+2 ) : 16 

The first term of our numerical series when smoothed is: 

di = (ag + 4aio + 6#i + 4a 2 + #3) : 16 
= (11 + 52 + 72 + 124 + 9): 16 = 17 

Numerical series should be smoothed only if absolutely neces- 
sary. The smoothing procedure can efface essential characteristics 
of the series and lead to false conclusions. 



CHAPTER V 
HARMONIC ANALYSIS 

V. 1. THE ANALYSIS 

THE HARMONIC ANALYSIS is one of the methods for describing 
periodic phenomena. The latter are indicated by a varia- 
tion in which the dependent variable shows a repetition at 
equal intervals of the other variant. A great number of climato- 
logical elements reach one extreme value at about low sun and 
the opposite extreme at about high sun. The periodical variation 
of the altitude of the sun is the physical cause of the periodical 
annual variation of these elements. Their daily course frequently 
offers a complex problem. The physical conditions during the day 
are different from those during the night. A sound physical stand- 
point should never be neglected in investigations of periodical 
phenomena. 

The harmonic analysis makes use^D/ Fourier's series. We start 
from the following formula, which is rather common in meteor- 
ological and climatological investigations: 

y = a<> + #1 sin (x + AI) + a* sin (2x + A%) + 

+ a* sin (ix + Ai) + 

where a is the arithmetic mean of the observations and #1, a 2 , a* 
are amplitudes (half ranges) of the superimposed waves. Ai, 
At, -Ai are called phase angles, determining the times at which 
the extremes occur. The meaning of the equation mentioned is 
easy to understand by means of the graph of Figure 15. 

The symbol x represents the time angle. It is computed from 

360 



if P denotes the length of the period investigated. For example, 
if P = 12 months, then 

* = TT = 30 ' 2x = 60 ' etc ' 

70 



HARMONIC ANALYSIS 



71 




30 60* 90* 120 150 180 2JO 240 270* 300* 330 360* 
FIG. 15. Example of superposition of two waves: 

I = aisin (x + Ai)\ AI = 
II = a 2 sin (2x + AJ)\ A 2 = 90 
III = sum of the waves I and II 



If P = 24 hours 



360 1 co 

x = -=-r- = 15 etc. 
24 



Further, it is easy to see that 

y = a + a\ sin x cos A i + a 2 sin 2x cos -42 + 
+ &i cos x sin .4 1 + a 2 cos 2# sin ^4 2 + 



We substitute : 



= a\ sn <r, 
= ai cos A\\ 



= ^2 sn 
=^2 cos 



Then, 

y = a + pi cos # + 2 cos 2x + + gi sin # + g 2 sin 2x + 

w , tf i, #i w n _i may be the deviations from the arith- 
metical mean and are inserted into the equation in place of y. 

Then we get n equations from which the constants pi, qi, 
pi, 32 can be calculated. 

If the number of observations is greater than that of the con- 
stants (the common case), the method of least squares is to be 
applied. This method yields the following equations for the 



72 METHODS IN CLIMATOLOGY 

constants : 

pi = 2/n[_u cos + #1 cos x 

+ Uz COS 2tf + ' ' ' ^n-l COS (ft 

w sin + u\ sin x 

-f* ^2 sin 2# -}- u n \ sin (w 

"w cos + HI cos 2x 

-|- ^2 cos 4# -[- w n _i cos 2 (ft 

sin + u\ sin 2x 

+ 2*2 sin 4# + w n -i sin 2 (ft 

o COS + Wi COS kx 

+ 2/2 cos 2&# + u n -i cos (ft 

t, sin + ^i sin kx 

+ w 2 sin 2^ + ^n-i sin (ft - 

According to the definitions of p k and g>, 

^ fc j A $* k 

Obviously, also, the following solutions are valid : 

L. W. Pollak J introduced an excellent procedure for evaluating 
the equations mentioned, which is so clear and simple that even 
those without special mathematical training are able to make the 
necessary calculations. 

In the following, therefore, Pollak's system is illustrated. 
For lack of space, only two special problems, frequently used in 
climatology, can be discussed : 

1) 12 equidistant observations (average values) 

2) 24 equidistant observations 

Case 1 usually represents the annual course of an element; 
case 2, the diurnal course. There is, however, no difficulty in 
calculating the constants of a diurnal course with 12 two-hourly 

1 L. W. Pollak, "Handweiser zur harmonischen Analyse/' Prager Geophysikalische 
Studien, vol. II (Prague, 1928); L. W. Pollak, Rechentafeln zur harmonischen Analyse, 
(Leipzig, 1926). I learn from a letter that Dr. Pollak is extending the Rechentafeln up 
to 100 ordinates. The first edition reached 40 ordinates. 



HARMONIC ANALYSIS 



73 



values and in evaluating an annual variation by means of 24 half- 
monthly means, if greater accuracy is required. 

Generally, the monthly means are related as was said above 
to the middle day of the month: e.g. the 14th, 15th, or 16th day of 

TABLE 17. INSTRUCTIONS FOR CALCULATING THE CONSTANTS FOR 12 EQUIDISTANT 
VALUES YIELDING THE FIRST, SECOND, AND THIRD TERM OF FOURIER'S SERIES 

(After L. W. Pollak) 



i 


Pi 


01 


P* 


0J 


P* 


01 





1 





1 





1 





1 


0.866 


0.5 


0.5 


0.866 





1 


2 


0.5 


0.866 


-0.5 


0.866 


-1 





3 





1 


-1 








-1 


4 


-0.5 


0.866 


-0.5 


-0.866 


1 





5 


-0.866 


0.5 


0.5 


-0.866 





1 


6 


t 





1 





-1 





7 


-0.866 


-0.5 


0.5 


0.866 





-1 


8 


-0.5 


-0.866 


-0.5 


0.866 


1 





9 





-1 


-1 








1 


10 


0.5 


-0.866 


-0.5 


-0.866 


-1 





11 


0.866 


-0.5 


0.5 


-0.866 





-1 



TABLE 18. THE SAME AS IN TABLE 17, BUT FOR 24 EQUIDISTANT VALUES 

(After L. W. Pollak) 



f 


Pi 


Qi 


1 


0* 


P* 








1 





1 





1 





1 


0.966 


0.259 


0.866 


0.5 


0.707 


0.707 


2 


0.866 


0.5 


0.5 


0.866 





1 


3 


0.707 


0.707 





1 


-0.707 


0.707 


4 


0.5 


0.866 


-0.5 


0.866 


-1 





5 


0.259 


0.966 


-0.866 


0.5 


-0.707 


-0.707 


6 





1 


i 








-1 


7 


-0.259 


0.966 


-0.866 


-0.5 


0.707 


-0.707 


8 


-0.5 


0.866 


-0.5 


-0.866 


1 





9 


-0.707 


0.707 





-1 


0.707 


0.707 


10 


-0.866 


0.5 


0.5 


-0.866 





1 


11 


-0.966 


0.259 


0.866 


-0.5 


-0.707 


0.707 


12 


-1 





1 





-1 





13 


-0.966 


-0.259 


0.866 


0.5 


-0.707 


-0.707 


14 


-0.866 


-0.5 


0.5 


0.866 





-1 


15 


-0.707 


-0.707 





1 


0.707 


-0.707 


16 


-0.5 


-0.866 


-0.5 


0.866 


1 





17 


-0.259 


-0.966 


-0.866 


0.5 


0.707 


0.707 


18 





< 


-1 








1 


19 


0.259 


-0.966 


-0.866 


-0.5 


-0.707 


0.707 


20 


0.5 


-0.866 


-0.5 


-0.866 


-1 





21 


0.707 


-0.707 





1 


-0.707 


-0.707 


22 


0.866 


-0.5 


0.5 


-0.866 





-1 


23 


0.966 


-0.259 


0.866 


-0.5 


0.707 


-0.707 



74 METHODS IN CLIMATOLOGY 

the month, depending on the length of the respective months. 
Hourly means are related to the 30th minute of the hour. 

Tables 17 and 18 contain instructions for calculating the con- 
stants 

Ply ffli 2, 22, p3, ?3, 

and from these, 

#1, #2, #3 and Ai 9 A% 9 Aa 

The first column of each table contains the number of the item. 
For example: the mean of May has the number i = 4, because 
January has the number zero. The value of 5 A.M. has number 
i = 5 ; if the series begins with midnight = hour. 

In columns p\, qi. p^ 2 the factors by which the respective 
items (see Column i} have to be multiplied, are indicated. 

The columns pi, qi, etc., are totaled (naturally, with regard to 
sign). 2 The sums multiplied by 2/n (that is, 1/6 for 12 values, and 
1/12 for 24 values) yield the constants pi, qi, etc. The multiplica- 
tions are done with the aid of a slide rule, a multiplication table, 
or a calculating machine. The best expedient is Pollak's Rechen- 
tafeln, which are indispensable if such calculations are made on 
greater scale, or if other lengths of period are considered. In 
these calculations, however, one shoufcl shun a purely arithmetical 
accuracy not guaranteed by the exactness of the observations. 
Generally, results to three places of decimals are ample. 

If the constants p and q are calculated, the amplitude a and the 
phase angle A, are easily computed with the formulae given above. 

Although the problem is exceedingly simple, blunders some- 
times happen in determining the true phase angle. The diagram 
(Fig. 16), helps to avoid such mistakes. Possible combinations 
of the signs of p and q for which the phase angle is to be determined, 
are: 

a) +P + q; 

b) + p-q\ 

c) - p - 2; 

d) - P + q 

1 The deviations u<>, ui, ,t/ n -i are multiplied with the factors of each of the 
columns p\, q\\ p*, g 2 ; . These series of products are entered into a new blank, with 
the same headings pi, q\\ p^ q^\ -. The column p\ of the new blank contains, thus, 
the following products (in the case of Table 17): 

1 X w 
0.866 X ui 



0.866 X ttn-i etc. 
These columns p\ t q\\ P*,qi\ are then totaled. 



HARMONIC ANALYSIS 
90 



75 



I80l 




270 

FIG. 16. Diagram for determining the true phase angle 

From these combinations, we calculate directly the following 
angles : 



a) A. 



b) 



c) 



d) 



Then the true phase angle A is : 



a) A = -<4++ A in the first quadrant 

b) A = 180 A+- Am the second quadrant 

c) A = 180 + A __ A in the third quadrant 

d) A = 360 A-+ A in the fourth quadrant 

The following example, 3 Table 19 shows each step of the calcu- 
lation, so that a few explanatory remarks are sufficient. 

Taken from V. Conrad, "Anomalien und Isanomalen der Sonnenscheindauer in 
den oesterreichischen Alpen," Beihefte Jahrbuch d. Zentralanstalt f. Meteorologie (Wien, 
1938); V. Conrad, "Die Komponenten der Jahresschwankung der Sonnenscheindauer/' 
Helvetia Physica Acta, vol XII (1939), p. 38. 



76 METHODS IN CLIMATOLOGY 

TABLE 19. HARMONIC ANALYSIS OF THE ANNUAL COURSE OF THE RELATIVE SUNSHINE 

DURATION (S in %) AT AN AVERAGE LEVEL OF 1000 FEET 

IN A MOUNTAINOUS COUNTRY 

(di = deviations) 



(1) 


(2) 


(3) 


(4) 


(5) 


(6) 


(7) 


(8) 


Month 5(%) 


i 


di 


Pi 


Qi 


P* 


q* 


I 


25 





-19 


- 19 





-19 





II 


39 


1 


- 5 


- 4.330 


- 2.5 


- 2.5 


- 4.330 


III 


47 


2 


+ 3 


+ 1.5 


+ 2.598 


- 1.5 


+ 2.598 


IV 


45 


3 


+ 1 





+ 1 


- 1 





V 


53 


4 


+ 9 


- 4.5 


+ 7.794 


- 4J5 


- 7.794 


VI 


58 


5 


+ 14 


- 12.124 


+ 7 


+ 7 


-12.124 


VII 


62 


6 


+ 18 


- 18 





+ 18 





VIII 


58 


7 


+ 14 


- 12.124 


- 7 


+ 7 


+ 12.124 


IX 


54 


8 


+ 10 


- 5 


- 8.660 


- 5 


+ 8.660 


X 


41 


9 


- 3 





+ 3 


+ 3 





XI 


29 


10 


-15 


- 7.5 


+ 12.990 


+ 7.5 


+ 12.990 


XII 


19 


11 


-25 


- 21.650 


+ 12.5 


-12.5 


+ 21.650 


mean 


(flo) 44 






-102.728 


+ 28.722 


- 3.5 


+33.774 


Pi 


- - 17.121 




kPi 


= 1.2335 pi 


= - 0.583 


lg/>2 


= 9.7657 


tfi 


= + 4.787 




lgi 


= 0.6801 q z 


= + 5.629 


lg$2 


= 0.7504 


(Ai) 


= 7422' 




Ig tan A i 


= 0.5534 (,4 2 ) 


= 555' 


Ig tan A* 


= 9.0153 


a\ 


= 17.8% 




kPi 


= 1.2335 . a 2 


= 5.7% 


lg/>2 


= 9.7657 


A, 


= 28538' 




Ig sin A i 


= 9.9836 w4 2 


= 3545' 


Ig sin A 2 


= 9.0132 



Ig ai = 1.2499 Ig a 2 = 0.7525 

S 44 + 17.8 sin (x + 286) + 5.7 sin (2* + 354) 

Column 2 contains the original numbers, the periodical trend 
of which is to be analyzed. Columns 5 to 8 contain the products 
of the consecutive deviations (see column 4), and of the values of 
the trigonometrical functions of the respective angles as indicated 
in Table 17. 

V. 2. EVALUATION OF THE EQUATION 
Table 20 gives a pattern of the evaluation of Fourier's series: 
5 = 44 + 17.8 sin (x + 286) + 5.7 sin (2* + 354) 

This evaluation is to be recommended, since it is the only check on 
the correctness of the analysis. The graph, Figure 17, furnishes a 
good illustration of the agreement of observation and calculation. 
The example shows, too, that the combination of only two waves 
is sufficient to give a good approximation with the rather complex 



HARMONIC ANALYSIS 

TABLE 20. EVALUATION OF EQUATION 5 IN TABLE 19 



77 



(1) 


(2) 


(3) 


(4) 


(5) 


(6) 


(7) 


(8) 


(9) 


Month 


* 


(x + A0 


2x -{-At 


18 sin (x -f -4i) 


6 sin (2* -|- A 


) C. 


O. 


o-c 


Jan. 





-7422' 


- 555' 


-17.1 


-0.6 


-18 


-19 


-1 


Feb. 


30 


-4422' 


+ 5405' 


-12.4 


+4.6 





- 5 


+3 


March 


60 


-1422' 


+6555' 


- 4.4 


+5.2 


+ i 


+ 3 


+2 


April 


90 


+ 1538' 


+ 555' 


+ 4.8 


+0.6 


+ 5 


+ 1 


-4 


May 


120 


+4538' 


-5405' 


+ 12.7 


-4.6 


+ 8 


+ 9 


+ 1 


June 


150 


+ 7538' 


-6555' 


+ 17.2 


-5.2 


+ 12 


+ 14 


+2 


July 


180 


+ 7422' 


- 555' 


+ 17.1 


-0.6 


+ 17 


+ 18 


+ 1 


Aug. 


210 


+4422' 


+ 5405' 


+ 12.4 


+4.6 


+ 17 


+ 14 


-3 


Sept. 


240 


+ 1422' 


+6555' 


+ 4.4 


+ 5.2 


+ 10 


+ 10 





Oct. 


270 


-1538' 


+ 555 ; 


- 4.8 


+0.6 


- 4 


- 3 


+ 1 


Nov. 


300 


-4538' 


-5405' 


-12.7 


-4.6 


-17 


-15 


+2 


Dec. 


330 


-7538' 


-6555' 


-17.2 


-5.2 


-22 


-25 


-3 
















2(+) 


- 12 
















s(-) 


- 11 


C. 


= Calculated deviations. 


O. 


= Observed deviations. 



I II III IV 



VI VII VIII IX 



XI XIJ I 



tio 
o 

-10 
-20 




+ 10 



-10 





FIG. 17. At the top: the annual course of the relative duration of sun- 
shine in a given average level of a mountainous region. Full line: calculated. 
Dots: observed. 

At the bottom: the two constituents of the calculated curve. 



78 METHODS IN CLIMATOLOGY 

observed curve. As far as the approximation is concerned, atten- 
tion should be directed to the convergence of the resulting series. 

The ratio - = 1/3, and - - 1/9 (a, not included in Table 19), 
di di 

so that here a good convergence exists. Thus, the series can be cut 
off at the second term, in the example of Table 19. 4 

V. 3. RELATIVE AMPLITUDES 

The amplitudes #1, a 2 , a 3 , are given in the units of the devia- 
tions. The meaning of the amplitudes is dependent upon the 
particular arithmetical mean. For purposes of comparison, the 
amplitudes of the different terms have to be freed from the in- 
fluence of the unit and that of the arithmetical mean: to do this, 
the amplitudes are divided by the arithmetical mean. 

d\ d% 



In the example (Table 19) : 

d R>l = 0.40 a*, 2 = 0.13 

t 
Often it is advantageous to give the relative amplitudes in per 

cent of the arithmetic mean. Then 

a Rt i = 40%, a*, 2 = 13% 

V. 5. TIMES OF THE EXTREMES 

The times of the extremes of the single constituent waves are 
easily calculated from the equation : 

kx + A k = 90 or 450 respectively, 

4 D. Brunt (Meteorological Magazine, 1937, p. 268), gives a criterion as to whether 
or not it pays to calculate further terms of the Fourier's series. Brunt's formula reads: 



where <r 2 means the variance of the value (di, column 4 in Table 19) subject to the analysis 
In the example 

ff . = 5*'=179 
n 

i(ai* + a, 8 ) 175 
Therefore 

a' 2 = 4 

This means that cr /2 is only 2% of a 2 . Hence the criterion indicates that further terms 
of the series can be neglected. 



HARMONIC ANALYSIS 79 

where k equals the number of the term. Example from Table 19: 

L ~~ t /Time of maximum = x + A\ = x + 286 = 450 or 
k = 1 J 

*ma* = 164 

* m in = 164 + 180 = 344 

Since 360 degrees correspond to an average year of 365.25 days, 
the angle-degrees have to be multiplied by 1.0145, if the conversion 
into days is required. The date is obtained by adding 16 to the 
product. This is clear, because the mean values of the elements 
are related to the midpoint of the month (generally January). 

Example: If x = 164, 1.0145 X 164 = 166 days have been 
passed between the middle of January and the time of the maxi- 
mum. 6 Therefore the maximum of the whole-year-wave occurs 
on the 166 + 16 = 182nd day of the year: that is, July I. 6 

It goes without saying that the starting date can be chosen 
arbitrarily. The hydrologist, for instance, begins his year on 
July first, October first, November first, or any other date repre- 
senting the beginning of the chief rainy season. 

The problem of calculating the time of the extremes directly 
from Fourier's series with two terms was simply solved nearly a 
century ago. The solution gives a good approximation. 7 Gen- 
erally, it is sufficient to take the times of the extremes from a good 
graph. In our example, the curve representing the annual course 
of the relative duration of sunshine (Fig. 17) indicates, approxi- 
mately, August 1 for the maximum and December 16 for the 
minimum. 

Harmonic analysis permits the interpolation of missing or 
dubious values of a periodical series and is an excellent means of 
investigation in many problems of periodicity. The example of 
Table 19 is taken, for instance, from an investigation in the varia- 
tion of the annual course of the duration of sunshine with altitude. 
By means of the harmonic analysis, it was possible to puzzle out 
the complex problem and to show among other results that 
the amplitude of the second harmonic term is invariant with height. 8 
In order to illustrate the high research value of the harmonic an- 

8 It is sufficient to increase the number of degrees by li%. * See Appendix IV. 

7 Hugo Meyer, Anleitung zur Bearbeitung meteor ologischer Beobachtungen (Berlin, 
1891), p. 38 ff. 

8 This result is important in so far as it indicates that, against every expectation, the 
half-year wave is the main wave upon which the whole-year wave of very variable am- 
plitude and phase angle is superposed. 



80 METHODS IN CLIMATOLOGY 

alysis, attention is drawn to Hann's excellent studies of the daily 
course of air pressure, as also to the numerous investigations which 
deal with the idea of Arthur Schuster's "expectancy/* and to the 
methods of L. W. Pollak and Sir Gilbert Walker. 

Even if the climatologist is not compelled de facto to use the 
harmonic analysis for his own work and investigations, he must 
understand this method, so often applied in scientific literature. 



PART II 

REPRESENTATION OF CHARACTERISTIC 

FEATURES OF DIFFERENT 

ELEMENTS 

IT is NEITHER desirable nor possible to give detailed instructions 
as to how to handle each element and its characteristics. The 
first part of this book has offered the mathematical means of 
descriptive statistical characteristics for every series of numbers, 
independent of the element. The only restriction already men- 
tioned was that here scalar quantities, not vectorial, were 
considered. 

Many of the common characteristics have been discussed pre- 
viously. A partial summary of the most usual characteristics is 
given in the forms of climatic tables in the Appendix. The follow- 
ing sections deal with a selection only of important and less well- 
known characteristics. 

CHAPTER VI 
TEMPERATURE 

VI. 1. HOURS OF OBSERVATION. REDUCTION 
TO THE "TRUE MEAN" 

THE INTERNATIONAL REGISTER assumes that the elements, 
with the exception of precipitation, are observed three 
times a day. This system is used in most of the European 
countries, in Russia, from the Baltic Sea to the east Asiatic coast, 
and in some other networks of climatological stations. The 
method of three daily observations is therefore important for the 
climatographies of at least two continents. The observations 
are so distributed over the day that one is made in the morning, 
one in the afternoon, and the third in the evening. 

For an easier understanding, Figure 18 represents a daily 
course of temperature derived from 50 years 1 thermograph records 
and related to the middle day of the month. The average tem- 

81 



82 



METHODS IN CLIMATOLOGY 



perature is characterized by the ordinate of that rectangle which is 
equal to the area enclosed by the curve (top) , the abscissa (bottom) , 
in Figure 18 located at 14C, and the limiting ordinates (sides). 
Generally this area is measured by a planimeter. In the case of 
a smooth curve, such as that in Figure 18, one can estimate the 



c 

22 

20 

M 
18 

16 
14 



HOURS 
II N I 3 



9 II M 



.M 





1.9 1.6 | I.2|0l8| 0.8(1.5 J2.9M.O 5Op.9 |68|Z5 |a9|e.7|a6 kelrsks |5.4|4.4| 3.TP3 \2.7 

AREAS, 0.2 5 SQ. CM 

t 

FIG. 18. The daily course of temperature at a place in 
Central Europe, in August (C) 



areas between the ordinates of two consecutive hours; the sizes 
of these areas can be read from the numbers on the J\T-axis, The 
unit is 0.25 sq cm. The sum of these areas divided by the number 
of intervals (i.e., 24) gives the height of the average ordinate. 
In the present case, 4.57C results. 

Therefore the average temperature required is 4.57 + 14.00 
= 18.57C. The method of numerical integration is correct but 
complicated. 

TABLE 21. AVERAGE DAILY COURSE OF TEMPERATURE AT 
A PLACE IN CENTRAL EUROPE (AUGUST) 



c 

6 



8 



10 



11 



12 



A.M. 16.1 15.8 15.4 15.1 14.8 15.0 16.1 17.4 18.5 19.4 20.3 21.1 
P.M. 21.9 22.6 22.7 22.5 21.9 21.0 19.8 18.8 18.1 17.5 17.0 16.6 

arithmetic mean 18.56 C 



TEMPERATURE 83 

We can arrive also at a practically identical result by a very 
simple computation. The arithmetic mean of the 24-hourty 
temperatures (see Table 21) yields 18.56C. 

Thus the arithmetic mean of the 24-hourly temperatures is 
defined as the true mean temperature. 1 This definition is valid 
for all elements which are continuously recorded or observed at 
each hour, except vectorial values. 

We return to the international form of the climatological 
register and observations at certain fixed hours of the day. The 
aim of these repeated observations during the day is to learn some- 
thing about the daily variation of temperature (or another ele- 
ment), on the one hand; and, on the other hand, to figure out the 
average temperature of an individual day, and the true means for 
the months and for the year. 

Two statistical results are immediately derived from the three 
(two in some networks) daily observations: (1) The average of the 
daily observations is defined as the daily mean (valid for all ele- 
ments except wind direction). (2) The temperatures of each ob- 
servation hour are averaged over the month. Monthly average 
temperatures for the fixed hours result. 

A simple combination of these average observational hours 
should yield the true monthly mean. A difficulty exists in so far 
as the form of the daily course is variable from region to region, 
and in the course of the year. 

Before we continue this discussion, it is necessary to give some 
numerical examples of observational hours in the United States 
and elsewhere. 

The hours which are highly recommended by the International 
Meteorological Committee 2 and used in many European meteor- 
ological networks are : 

7 A.M., 2 P.M., 9 P.M. 

1 The usual nomenclature is true daily mean. This denomination is appropriate as 
long as 24-hourly temperatures are concerned. Then the expression is valid for any 
single day as well as for the average of the month. It is another matter with observa- 
tions at fixed hours. We then have to discriminate between: (a) the daily mean and (b) 
the mean of the monthly averages of the temperatures at fixed hours. In case (a), we 
speak only of a daily mean, which will be discussed later on. Only from the monthly 
average temperatures at the fixed hours can a combined and corrected value be derived 
which shows an optimum of agreement with the 24-hour mean. This approximation is 
called true mean in the following discussion. 

1 International Meteorological Codex, p. 24. 



84 



METHODS IN CLIMATOLOGY 



Other recommended hours are : 

6 A.M., 2 P.M., 10 P.M. 

7 A.M., 2 P.M., 10 P.M. 

7 A.M., 1 P.M., 9 P.M. 

8 A.M., 2 P.M., 8 P.M. 3 

In the United States the hours of observations shown in Table 
22 were used. 4 

TABLE 22. OBSERVATION-HOURS IN THE UNITED STATES (AFTER FRANK BIGELOW) 



1870, Nov. 


to 


1872, 


Aug. 24, 


7.35 


A. 


M. 4.35 


P.M., 


11.35 


P.M. 


WMT> 


1872, Aug. 24, 


to 


1879, 


Oct. 31, 


7.35 


A. 


M. 


4.35 


P.M., 


11.00 


P.M. 


WMT 


1879, Nov. 1, 


to 


1886, 


Dec. 31, 


7.00 


A. 


M. 


3.00 


P.M., 


11.00 


P.M. 


WMT 


1887, Jan. 1, 


to 


1888, 


June 30, 


7.00 


A. 


M. 


3.00 


P.M., 


10.00 


P.M. 


75th MT 


1888, July 1, 


till 


about 1930 


8.00 


A 


M. 






8.00 


P.M. 


75th MT 


1930 to 1936, 








7,00 


A. 


M. 






7.00 


P.M. 




From 1936, 








7.00 


A. 


M. 


1.00 


P.M., 


7.00 P.M., 


1.00 A.M. 



Dr. C. F. Brooks has been so kind as to bring down to date Bigelow's data. 

6 WMT = Mean time of the meridian of the observatory at Washington, D. C. 
75th MT = Mean time of the meridian 75W = Eastern Standard Time. 
WMT - 75th MT 8.3 minutes. 

JP 

Table 23 contains examples of the annual course of the differ- 
ence between the temperature obtained from the combination of 
the observational hours (t ) and the true mean (derived from 24- 
hourly observations, (/*). The difference 

c = t t t 

is called correction to the "true mean.'* It is 

t t = t + c 

This means that the correction c is to be added to t without 
changing its sign. 

1 There is also a series of only two observational hours, recommended by the IMC: 

8 A.M., 8 P.M. 

9 A.M., 9 P.M. 
10 A.M., 10 P.M. 

The averages of the temperatures at these hours do not show great differences from the 
true mean but do not indicate anything about the daily variation of temperature. 

4 According to the investigations in this problem by Frank H. Bigelow, " Report on 
the Temperatures and Vapor Tensions of the United States," U. S. Dept. of Agriculture, 
Weather Bureau, Bulletin S (Washington, 1909). See also Alexander McAdie, Mean 
Temperatures and Their Corrections in the United States (U. S. War Dept., Washington, 
1891). 



TEMPERATURE 



85 



TABLE 23. THE ANNUAL COURSE OF "c" (F) FOR 
DIFFERENT HOURS OF OBSERVATION 



Buffalo, N. Y. (42.9N, 78.9W, 770 ft.) 


Central Europe 


Hours 


7A.M. 
3 P.M. 
11 P.M. 


8 A.M. 
8 P.M. 


max. 
min. 


7 A.M. 

2 P.M. 
9 P.M. 


7 A.M. 
2 P.M. 

2X9 P.M. 


Jan. 


0.0 


+ 0.4 


+0.4 


-0.9 


+0.4 


Feb. 


0.0 


+0.6 


+0.4 


-0.5 


-0.2 


Mar. 


+0.1 


+0.8 


+0.2 


-0.3 


+0.1 


Apr. 


+0.2 


+0.5 


-0.2 


-0.4 


-0.1 


May 


+0.3 


+0.2 


+0.1 


-0.7 


-0.3 


June 


+0.3 


0.0 


+0.3 


-0.9 


-0.5 


July 


+0.3 


0.0 


+0.3 


-0.9 


-0.5 


Aug. 


+0.3 


+0.4 


+0.3 


-0.1 


+0.3 


Sept. 


+0.4 


+0.8 


+0.4 


-0.1 


+0.3 


Oct. 


+0.3 


+ 1.2 


+0.5 


-0.4 


0.0 


Nov. 


+0.1 


+0.6 


+0.3 


-0.8 


-0.3 


Dec. 


0.0 


+0.4 


+0.5 


-1.0 


-0.7 


Year 


+0.2 


+0.5 


+0.3 


-0.6 


-0.2 


X\c\ 


2.3 


5.9 


3.9 


7.0 


3.7 


AR 


0<4 


1.2 


0.7 


0.9 


1.1 


ARx2\c\ 


0.9 


7.1 


2.7 


6.3 


4.1 



The first column shows c for three daily observations, the second 
for two, the third for the extremes, the fourth for three other hours, 
and the fifth offers a new feature. Here the observations at the 
three different hours are not combined in a common arithmetic 
mean, but in a weighted mean. Double weight is given to the 
average temperature at 9 P.M. Later on, we shall discuss this 
method. The question now to be answered is, which hours and 
which combinations of observations are most advantageous. 
There are two requirements: 

1) The sum of the absolute values of the monthly correc- 
tions should be as small as possible. 

2) The annual variation of the corrections should be as 
small as possible. 

The two requirements can be examined by simple criteria as 
follows: the third line from the bottom in Table 23 shows the sums 
of the monthly corrections, disregarding the signs (S|c|). It is 
clear that the first requirement is the better fulfilled the smaller the 
term S I c \ is. 



86 METHODS IN CLIMATOLOGY 

The second line from the bottom (Table 23) contains the an- 
nual range (AK) of the corrections, i.e., the difference between the 
largest and the smallest correction in the course of the year. It is 
obvious that the product of the two quantities should be as small 
as possible: 

AR X S|c| = minimum 

The hours 7 A.M., 3 P.M., 11 P.M., are indeed by far the best 
choice, and the hours 8 A.M., 8 P.M., the worst of the five. We see, 
too, that the result from identical hours of observation can be 
improved by weighted averages. The third combination (7 A.M., 
2 P.M., 9 P.M.) is only slightly better than the second) (8 A.M. 
+ 8 P.M.). The weighted mean is much better than either. (See 
last line of Table 23.) Nevertheless, three observations a day are 
much more advantageous, since they give rather good information 
about the daily variation of temperature. That is not the case at 
8 A.M. and 8 P.M., because these times are close to the hours of the 
daily mean. 

The difference between the average of the hours of observations 
and the true mean can be decreased by giving different weight to 
the observations as has been shown in one special example. If 
we assume three observations a day, then generally, the following 
equation holds: 

I = pta + qtb + rt c 

if i stands for "true mean," the subscripts a, b, c denote the 
different hours of observations; the letter /, the average monthly 
temperature at the respective hours. The letters p, q, r are coeffi- 
cients which have to be computed from the observations. Ob- 
viously the condition is valid : 

p+q+r = 1 

If there are complete observations at the fixed hours and syn- 
chronous, continuous records of temperature, say for two Januaries, 
the problem is very simple. 6 

There are li and 2 from the continuous records and / a ,i, t a ,*, 
/&,i *&,2, *c,i *c,2 for the average temperatures at the observational 
hours in the two January months, indicated by the subscripts 1, 2. 

9 The once chosen hours and the resulting combination of the monthly mean yields 
variable differences from the "true mean" in the course of the year. 



TEMPERATURE 87 

Then the following equations are valid : 

pta.i + 2/6,1 + rt Ct i = li 

plat + 3^6,2 + r/a,2 = *2 

p + q + r = 1 

From these three equations, the coefficients p, q, r are computed ; 
in reality more than 2 years' observations have to be available, in 
which case the three coefficients can be calculated with the method 
of least squares. 6 

Examples of weighted arithmetic means of the temperatures at 
fixed hours of observation follow: 

a) For the greater portion of Europe the observational hours 
7 A.M., 2 P.M., 9 P.M., with double weight given the 9 P.M. tempera- 
ture, are very advantageous: 

| = (/ 7 + / 2 + 2/ 9 ) : 4 = 0.25 t 7 + 0.25 / 2 + 0.50 / 9 , or 

p = 0.25, q = 0.25, r = 0.50 

b) Subtropics: 

I = D? + fe + /9 - 1/10 (/i - /Q)]: 3, or 

= 0.33, q = 0.30, r = 0.37 

c) Tropics: 

I = [2(* 7 + /*) + 3/9]: 7, or 

p = 0.286, q = 0.286, r = 0.428 

The same method is applicable if only the extremes of tempera- 
ture are observed once a day. 

This problem is of special importance. About 5500 coopera- 
tive observers in the United States make but one observation 
within the 24 hours of a day, and the same method is used in 
other countries. 

For reducing the extreme temperatures to the true mean the 
formula used is: 

t t = m + k(M - m) 

For further information see Nils Ekholm, "Reduction of Air Temperatures at 
Swedish Stations to a True Mean/' MWR, vol. 45 (1917), p. 58; Nils Ekholm, "Calcul 
de la temperature moyenne de 1'air aux stations mtorologiques Su^doises," Appendicc 
aux observations mttiorologiques Sutdoises, vol. 56, 1914 (Stockholm 1916). C. E. P. 
Brooks, "True Mean Temperature," MWR, vol. 49 (1921), p. 226. 

For instruction for the application of the method of "least squares," see, for ex- 
ample, H. Arkin and R. R. Colton, An Outline of Statistical Method* (4th ed., New York, 
1939). 



88 METHODS IN CLIMATOLOGY 

where M and m stand for maximum and minimum, and k is a 
factor which is more or less constant for a rather large region and 
is valid for a certain month ; k varies in the course of the year. 

C. E. P. Brooks 7 studied this problem in a general manner by 
calculating a correction term which reduces the arithmetic mean 
of M and m to the true mean. His formula reads: 



where a, b, c, are coefficients and R = M m. 

The correction term depends principally upon the average 
aperiodical range of temperature (R). It is variable in the course 
of the year because R has an accentuated annual course at least 
within the continents. 

The problem is complicated, furthermore, because the coeffi- 
cients a, b, c, are functions of the altitude (h) of the place above 
sea level. It is 

a = - 0.30 + 0.14 h 
b = 0.00 - 0.07 h 
c = - 0.0034 + 0.006 h 

The elevation h is expressed in kilometers, and the temperatures 
are absolute temperatures, i.e., T = tC + 273C. 
Example of a computation of the correction term: 

Blue Hill (42.2N. 71.1W, 640 ft.) 

July: M = 79F, m = 61F, h = 640 ft. 

converted into absolute temperature and kilometer respectively : 

M = 299 abs., m = 289 abs., h = 0.2 km 

Therefore: a = - 0.27, b = - 0.01, c = - 0.0022, R = 10C 



Correction term = a + bR + 

= - 0.272 - 0.140 - 0.220 = - 0.632 

The true mean is therefore 0.6 C or 1.1 F lower than the arith- 
metic mean of the extremes at Blue Hill, Mass., in the July average. 
Bigelow determined the difference in question empirically and 
got 0.5 F. Thus the agreement is not very good as far as the 
example at random is concerned. 

7 C. E. P. Brooks, "The Reduction of Temperature Observations to Mean of 24 
Hours, and the Elucidation of the Diurnal Variation in the Continent of Africa/ 1 
Quarterly Journal of the Royal Meteorological Society, London, vol. 43 (1917), pp. 375-387, 



TEMPERATURE 89 

The corrections derived from the formula mentioned above are 
tabulated in Table 24 for different altitudes between km and 
3.5 km and for different average aperiodical ranges between 4C 
and 20C. 

TABLE 24. CORRECTION REQUIRED TO REDUCE THE MEAN OF THE EXTREMES (M t m). 

DIFFERENT AVERAGE NON-PERIODIC DAILY RANGES R (C) = M - m AND 

DIFFERENT ELEVATIONS (h t METERS) ARE THE ENTRIES. 

(AFTER C. E. P. BROOKS) 



K(C)\ 500 1000 1500 2000 2500 3000 3500 



4 


-0.35 


-0.42 


-0.49 


-0.56 


-0.63 


-0.70 


-0.77 


-0.84 


6 


-0.42 


-0.50 


-0.58 


-0.66 


-0.74 


-0.82 


-0.90 


-0.98 


8 


-0.52 


-0.59 


-0.65 


-0.72 


-0.78 


-0.85 


-0.91 


-0.98 


10 


-0.61 


-0.67 


-0.70 


-0.73 


-0.76 


-0.79 


-0.82 


-0.85 


12 


-0.79 


-0.76 


-0.73 


-0.70 


-0.67 


-0.64 


-0.61 


-0.58 


14 


-0.97 


-0.85 


-0.73 


-0.61 


-0.49 


-0.37 


-0.25 


-0.13 


16 


-1.17 


-0.94 


-0.71 


-0.48 


-0.25 


-0.02 


+0.21 


+0.44 


18 


-1.40 


-1.04 


-0.68 


-0.32 


+0.04 


+0.40 


+0.76 


+ 1.13 


20 


-1.66 


-1.14 


-0.62 


-0.10 


+0.42 


+0.94 


+ 1.46 


+ 1.98 



Frank H. Bigelow made another step in his investigation. 8 
When he had computed the annual course of the correction of 
(M + m)/2 to obtain the true mean, he represented the monthly 
values cartographically and gave isolines of equal corrections for 
the United States. The result is the exact and concise summary of 
an immense numerical investigation which is so instructive that at 
least the two maps of January and July may be offered here. 
(Figs. 19 and 20.) 

The maps show that the trend of the average daily course of 
temperature is variable from region to region over the continent, 
as do also the corrections, which give the true mean. It must be 
emphasized that these variations are not so insignificant. In 
January they reach 1 F and in July, 2 F. 

For further purposes, it is important to remember this fact: 
Mean temperatures derived from extremes without corrections 
can show differences up to 2 Fahrenheit degrees at places in the 
United States, even if the true means for an individual month 
were identical. 

Some observatories and meteorological stations attempt to 
make more than three observations between dawn and dusk in 
order to get the closest approximation to the true mean. From a 

8 United States Department of Agriculture, Weather Bureau, Bulletin S (Washing- 
ton, 1909). 



90 



METHODS IN CLIMATOLOGY 




FIG. 19. Correction (F) to give the "true mean" if (M + m)/2 is known. 

Isolines for the United States, January. 

(After F. H. Bigelow) 




FIG. 20. The same as in Fig. 19, but for the month of July 



TEMPERATURE 91 

first glance at the curve of Figure 18 it is clear that rather the 
contrary is achieved, because the hours above the daily mean get 
more and more preponderance. 

If the hours of observation are changed in the course of time, 
or if the daily extreme temperatures are used instead of fixed hours 
of observations, the corrections vary in a discontinuous manner 
at the dates of changing. This can be a serious source of error, 
for instance in calculating periodicities. 

The corrections to the plain arithmetic mean vary up to a 
maximum of about 3/2 F in the United States, in July, if the hours 
7 A.M., 3 P.M., 11 P.M. are used, and reach about 2 F in the case 
of the arithmetic mean of the extremes. Therefore, the climatolo- 
gist who does not know whether the arithmetic means are weighted, 
corrected, etc., cannot compare the temperatures at different 
places with one another; or, if he does so, he should know that the 
accuracy of the comparison is less than perhaps 2 F. This 
reason alone makes it uneconomic to aggravate climatological 
computations, tables, and graphs, with decimals of degrees Fahren- 
heit. One exception may be a daily or annual course of tempera- 
ture derived from some years of observations. 

To avoid misunderstandings, let us repeat that all that has 
been told about the different corrections of the arithmetic means 
of temperatures at fixed hours of observations, etc., is valid only 
for averages of at least ten days. Generally these conclusions are 
applied only to the observation series of a month and longer 
periods. The average temperature of an individual day is defined 
as the arithmetic mean of the temperatures at the observation 
hours, or of the extremes. These averages can differ from the 
24 hours' mean of the day by a considerable amount. 

The foregoing discussion on the reduction of the arithmetic 
mean of the records at fixed hours of observation to the true mean 
is related to the readings of the dry-bulb thermometer. As far 
as the other climatological elements are concerned, such as vapor 
pressure, relative humidity, cloudiness, wind velocity, etc., it is 
usual to identify the arithmetic mean with the true mean. 

This is the common practice, but it does not claim that there 
would not be a better approximation to the true mean. For the 
greater part of the elements, investigations are lacking in this 
direction. 9 

9 The records of cloudiness at the hours of observation, 7 A.M., 2 P.M. (1 P.M.), 
9 P.M., are examined because of the differences from the true mean. (See V. Conrad, 
Met. Zeit. t 1928, p. 23.) It can be shown that the corrections for the above hours are 
not significant and can be neglected. 



92 



METHODS IN CLIMATOLOGY 



VI. 2. DATES AT WHICH THE AVERAGE TEMPERATURE 
CROSSES CERTAIN THRESHOLDS 

Generally, these dates are taken from graphs (drawn on a large 
scale) which represent the annual course. As this method is 
laborious, a simple interpolation is indicated here. The formula is 
founded on the fact that the trend of the ascending and descending 
branches of the curve is linear in rough approximation. There- 
fore, the interpolation method ife no longer applicable near the 
extremes. 



UP 



DOWN 



FIG. 21. Diagram explaining the interpolation of the dates at which tempera- 
ture crosses a given threshold upward and downward. (After V. Conrad) 

The symbol t means the given threshold of temperature, a 
the monthly average above it, and b that next below it. The 
letter D is the difference in days between the middles of the months 
with the average temperature (b or a) and the middle of the 
consecutive month (see Fig. 21). The symbol d up is the difference 
in days between the middle of the month below the threshold and 
the date at which the temperature (/ ) of the threshold occurs. 

According to G. Crestani ("Alcuni considerazioni sul calcolo della media della 
velocitA del vento, sull'andamento diurno ed anno della medesima in Italia," Boll, 
bimens. XLIX, 1930, 10-14), the combination of wind velocities, i'(9 A.M. + 3 P.M. 
+ 9 P.M.), yields corrections to the true mean which are zero in February and November 
and reach their maximum in May with about 7% of the true mean. On the average 
for the year, the correction is 4% of the true mean. 

If the velocity of the wind is only estimated, not instrumentally measured, the 
arithmetic mean of the above-mentioned observation hours shows negligible corrections 
to the true mean. Probably the same is valid with three other reasonable hours of 
observation. 



TEMPERATURE 93 

The symbol ddo is the same for the descending branch of the curve. 
Then: 

d up :D = (t - b): (a - 6) 
and 

d do :D = (a - t ):(a - b) 
Hence 



a b 
= D 



a t 



a 
It is sufficiently accurate to assume 

D = 30 days 
Finally the interpolation formula reads: 



u P o 

a b a b 

As an example, the annual course of temperature at Bismarck, 
N. D. (Table 25), will serve. 

TABLE 25. THE ANNUAL COURSE OF TEMPERATURE AT BISMARCK, N. D. 
(46.8N, 100.6W, 1670 FEET) 





O F (*) 


Deviations 
op 


F-Jan. 


Rel. Temp., % 


Jan. 


8.6 


-32.0 


0.0 





Feb. 


9.9 


-30.7 


1.3 


2 


Mar. 


24.3 


-16.3 


15.7 


26 


Apr. 


43.3 


+ 2.7 


34.7 


56 


May 


54.3 


+ 13.7 


45.7 


74 


June 


64.2 


4-23.6 


55.6 


91 


July 


70,0 


-f29.4 


61.4 


100 


Aug. 


67.5 


4-26.9 


58.9 


96 


Sept. 


57.9 


4-17.3 


49.3 


80 


Oct. 


44.6 


4- 4.0 


36.0 


59 


Nov. 


28.0 


-12.6 


19.4 


32 


Dec. 


14.7 


-25.9 


6.1 


10 


Year 


40.6 









* After R. DeCourcy Ward and C. F. Brooks, "The Climates of North America," 
Ktfppen-Geiger, Handbuch der Klimatologie, pt. J (Berlin, 1936). 

The dates when the temperature rises above and drops below 
32, 43, 65 should be calculated. The threshold of 32 divides 
the curve into the parts below and above the freezing point; 43 is 



94 METHODS IN CLIMATOLOGY 

about the temperature limit of germination of seeds, 65 is the 
assumed temperature limit for artificial heating. 

- n 32 -24.3 
up 43.3-24.3 

and 

dd - 3Q ^ - 32 23 
^*- JU 44.6 -28.0 -" 

rf up == 12 is to be added to March 16 and 
d do = 23 to October 16 

Thus we see that : 

The temperature rises above 32: March 28 
The temperature drops below 32: November 8 

In the same way it follows that : 

The temperature rises above 43: April 15 
The temperature rises above 65: June 19 
The temperature drops below 65: August 24 
The temperature drops b^low 43: October 19 

VI. 3. DURATION OF TEMPERATURES ABOVE AND 
BELOW CERTAIN THRESHOLDS 

These data are very characteristic for the annual course of 
temperature and should not be omitted. 

If a table is at hand with running numbers of the dates from 
January first, the calculation is nothing but a simple subtraction. 10 
The duration in days for the three thresholds mentioned, at Bis- 
marck, N. D. (see VI, 2), is: 

above 32F: 225 days = 32 weeks = 62% of the year 
above 43F: 187 days = 27 weeks = 51% of the year 
above 65F: 66 days = 9 weeks = 18% of the year 

Sometimes the conversion of the number of days into weeks can 
be recommended, if the observations are not very accurate, as 
also when a quick and easy survey is given. This purpose can be 
better attained with the expression that yields the smaller numbers. 
Often also the knowledge of the percentage of the year with tem- 
peratures above and below the thresholds is useful. 

10 See Annex IV. 



TEMPERATURE 95 

It is instructive to learn that at Bismarck, N. D., the average 
temperature remains below the freezing point for 38% of the year 
and above for 62%, or that only 18% of the year has a temperature 
above the technically assumed threshold for artificial heating. 

The duration of a temperature higher than 43 F is often used in 
climatological investigations and is called the vegetative period. In 
the United States the term growing season is more or less identical 
with "vegetative period. " It means the space of time between 
the last and the first killing frost, or, in the absence of satisfactory 
frost data, between the last and first minima of 32F or lower. 

Because I could not find a proper definition of "killing frost" 
in the available literature, Dr. C. F. Brooks was kind enough to 
give me the following information. I quote: "A killing frost is 
one which kills the general vegetation at a place. There is no 
specific meteorological definition. Where the vegetation is succu- 
lent and easily damaged by frost, a killing frost will occur at a 
higher temperature than where the vegetation is hardy. In the 
absence of killable vegetation, a temperature of 32F in a thermom- 
eter shelter is taken as a killing frost. Generally, however, the 
killing frost, first in fall and last in spring, occurs with a minimum 
shelter temperature higher than 32, sometimes as high as 40F." 

Therefore, "killing frost" and "growing season" are generally 
defined biologically, not quantitatively. 11 

It goes without saying that the principle of duration is ap- 
plicable also to many other climatological elements, as average 
duration of a certain wind direction, of the rainy or hot period of 
the year (see Figs. 40-41), of the snow-cover, of any average state 
of the atmosphere or of the surface of the ground, as far as period- 
ical climatological phenomena are concerned. In all these cases, 
any information about the respective durations is a good contribu- 
tion to the knowledge of the phenomenon. 

VI. 4. OTHER CHARACTERISTICS OF TEMPERATURE CONDITIONS. 
SPELLS OF COLD AND HOT DAYS 

Certain thresholds and definitions must be mentioned. In 
order to discriminate between (a) ice days and (b) frost days, "ice 
days" are defined as the days with the maximum below the freez- 
ing point, while "frost days" have the minimum below the freezing 
point. 

"See also W. G. Reed, "Frost and the Growing Season," in Atlas of American 
Agriculture (U. S. Dept. Agr., Washington, D. C., 1918). 



96 METHODS IN CLIMATOLOGY 

Days with a maximum ^ 77F are called (c) summer days, and 
days with a maximum ^ 86F are called (d) tropical days. 

All these thresholds are arbitrary. But they are used fre- 
quently and should be chosen in any case; others can be added 
which appear advantageous for special purposes. For such thresh- 
olds, frequencies should be computed. On the other hand, the 
thresholds are used for calculating spells of frost days, ice days, 
summer days, tropical days, etc. 

If some years' observations are at hand, the lengths of the 
respective spells are evaluated for each January, February, etc., 
of the series. Then, the average length and the greatest length of 
the spells should be given. 

If we have to handle ice days, summer days, tropical days, the 
temperature-maximum of the consecutive days is decisive for the 
length of the spells. In the case of frost days, it is the minimum 
that is decisive. 

In other investigations, the daily mean of consecutive days is 
considered. As far as the technique of counting out the duration 
of spells is concerned, the following rule is valid: If the spell begins 
in one month and ends in the next, the whole length in days is 
added to that month to which the greater part belongs ; if the spell 
is longer than two consecutive montfis, the two months have to be 
joined. 

VI. 5. CUMULATED TEMPERATURES 

Starting from a certain threshold, the mean daily temperatures 
above or below it are added together for the single month, for the 
season, or for the year. If some years' observations are available, 
the respective sums are averaged for the whole period. 

Especially in the earlier agricultural, meteorological, and 
climatological investigations, the mean daily temperatures above 
43 F (the temperature of germination as stated earlier) are totaled 
for the single month or for the growing season, etc. 

A. Angot calculated totals of daily minima at or below the 
freezing point. Table 26 gives these sums for the region of western 

TABLE 26. AVERAGE CUMULATED DAILY MINIMUM TEMPERATURES i 0C FOR PARIS, 
WINTER 1872/73 TO 1911/12 (AFTER A. ANGOT) 

Centigrade degrees 
Oct. Nov. Dec. Jan. Feb. Mar. Apr. May Year 

3.3 16.1 51.7 59.8 42.7 21.9 3.0 0.2 198,7 



TEMPERATURE 97 

Europe which is characterized by a climate of slight continentality. 
This procedure permits also a quantitative distinction between 
severe and mild winters, if a classification is made according to the 
standard deviation of the annual sums of minimum temperatures 
SOC. (See III, 3.) 12 

VI. 6. DEGREE DAYS 

The threshold for artificial heating is a daily mean of 65 F, 
according to the experience of heating engineers. If the daily 
mean drops below this temperature, heating is necessary and the 
amount of fuel to maintain a comfortable inside temperature is 
more or less intimately correlated with the number of degrees 
below 65F. This difference (65F minus each day's mean) is 
called number of degree days. The degree days are totaled for the 
month, the season, and the year, counting negative differences as 
zero. If a series of some years' observation is available, the cumu- 
lated degree days of the month, etc., are averaged. 

TABLE 27. DEGREE DAYS (F) AT BOSTON, MASS. (42.4N, 7l.lW, 125 FT.) 

AND CHICAGO, ILL. (41.9N, 87.6E, 673 FT.) 
(Boston 1914/15 to 1942/43; Chicago 1871 to 1936) 





Boston 


Chicago 


Jan. 


1101 


1256 


Feb. 


1027 


1083 


Mar. 


852 


903 


Apr. 


538 


545 


May 


248 


280 


June 


66 


69 


July 


8 


8 


Aug. 


16 


11 


Sept. 


98 


94 


Oct. 


338 


363 


Nov. 


651 


742 


Dec. 


1000 


1112 



Year 5943 6466 



Table 27 gives the cumulated degree days for Boston, Mass., 13 
and Chicago, 111. 14 The numbers for Boston are derived from 28 

12 See V. Conrad, " Die klimatologischen Elemente und ihre Abhangigkeit von ter- 
restrischen Einfliissen, Koppen-Geiger, Handbuchder Klimatologie, vol. I B (Berlin, 1936), 
p. 112. 

18 1 have to thank Mr. Edward Sable of the U. S. Weather Bureau for the Boston 
data. 

14 Taken from Thomas A. Blair, Climatology (New York, 1942), p. 18. 



98 



METHODS IN CLIMATOLOGY 



heating seasons (1914/15 to 1942/43), those for Chicago from the 
period 1871 to 1936. The comparison between the two series may 
be interesting. 18 




M J J 
S A 



FIG. 22. Relative temperatures at Bismarck, N. D. 

VI. 7. RELATIVE TEMPERATURES 

The comparison of the annual course of temperature at one 
place with that at another place presents two difficulties: 

16 In spite of unequal periods. 



TEMPERATURE 99 

1) The average temperatures of the year are different. 

2) The amplitudes (annual ranges) are different. 16 

The first inequality is overcome by calculating deviations from 
the arithmetic mean (annual average). (See Table 25, deviations.) 
The first and second are eliminated by the method of relative tem- 
peratures, a concept introduced by W. Koppen. The degrees by 
which the consecutive monthly averages exceed the temperature 
of the coldest month are each expressed in per cent of the difference 
between the average temperatures of the warmest and coldest 
months. By this explanation, the calculation of the numbers of 
the last column in Table 25 is easily understood. 

The author has given a graphical representation of relative 
temperatures 17 which facilitates the comparison of different series 
of relative temperatures. These are plotted at equal intervals to 
the right for the first seven months (Fig. 22). 

Starting from July, the curve turns back to the left, showing 
the values of the other months. Every point bears the initial 
of the month. Furthermore, the curve of relative temperatures 
for January through July is a full line, while the curve of the other 
half of the year is represented by a broken line. 

The asymmetry of the trend of the annual course of temperature 
at Bismarck, N. D., is evident. The months after July are warmer 
than the symmetrical ones before July. August is warmer than 
June, September warmer than May, and so forth. In this and in 
other respects, the representation in Figure 22 gives a good analysis 
of the annual curve, free from the embarrassing influence of the 
annual average and of the size of annual variation. 

16 The reader should distinguish between annual range and amplitude. The latter 
is half of the periodical range. 

17 V. Conrad, Fundamentals of Physical Climatology, p. 28. 



CHAPTER VII 

ATMOSPHERIC PRESSURE AND PRESSURE 
OF WATER VAPOR 

PRESSURES are statistically discussed in the same way as tem- 
perature. It should be emphasized that water-vapor pres- 
sure is a unilaterally limited element, and there are vast 
regions where the vapor pressure is practically zero in winter. 
It is not the aim of this book to deal with the different physical 
definitions regarding the water-vapor content of the atmosphere. 
For this, see V. Conrad, Fundamentals of Physical Climatology, and 
for a more detailed treatment, Sir Napier Shaw, Manual of 
Meteorology. 



100 



CHAPTERlVIII 
WIND 

VIII. 1. RESULTANT WIND VELOCITY CALCULATED FROM 
RECORDS OF DIRECTION AND VELOCITY 

THE RECORDS of an anemograph yield an average direction of 
the wind for each hour, and the run of the wind, in respec- 
tive units. 

The example, Table 28, shows the statistics derived from con- 
tinuous wind records at Batavia, Java, in January 1926. The 
first line F gives the number of hours with the various directions 

TABLE 28. EXAMPLE OF THE CALCULATION OF THE WIND-ELEMENTS; BATAVIA JAVA 
(6.2S, 106.8E, 26 FT.), JANUARY 1926* 







N 


NE 


E 


SE 


s 


sw 


w 


NW 


I 


Number of hours (F) 


153 


37 


1 


8 


41 


25 


92 


137 


II 


Run of wind 


309 


56 


(1) 


4 


29 


29 


176 


465 




(miles) (L) 


















III 


Average velocity 


2.01 


1.52 


(1.34) 


0.56 


0.72 


1.16 


1.92 


3.40 




(0 = L/F) (mi/h) 



















* Small differences of against the ratios L/F originate from the conversion of 
meters per second (m/s) into mi/h. 

during the period in question. The second line shows the run of 
the wind L for each direction during the same period (here the 
month of January). In the third line, the ratio L/F appears, i.e., 
the average velocities of the respective directions. The resultant 
run of the wind is calculated by Lambert's formula. 1 

C N = N - 5 + (NE + NW - SW - SE) cos 45 
C w = W - E + (NW + SW - NE - SE) cos 45 

where CN and Cw are the total run of the wind from north to south 
(subscript N) and from west to east (subscript W). If places only 
in temperate latitudes of the northern hemisphere are considered, 
it is recommended that the positive sign be given to winds from 
the north and from the west. It is practical and sufficiently exact 

1 Auxiliary tables for the use of this formula are given in the Smithsonian Tables. 

101 



102 METHODS IN CLIMATOLOGY 

to make the computation with the value cos 45 = 0.7, so that 
the calculation can be worked out mentally. 

C N = N - S + 0.7 (NE + NW - SE - SW) 
C w = W - S + 0.7 (NW + SW - NE - SE) 

For 16 directions we get: 

C N = N - S + (NNE + NNW - SSE - SSJF) cos 22 
+ (NE + NW - SE - 6W) cos 45 
+ (ENE + WNW - ESE - WSWQ cos 67| 



CV = W - E + (WWW + WSW - ENE - ESE) cos 

+ (NW + SW - NE - SE) cos 45 
+ (NNW + SSW - NNE - SSE) cos 67 

(The approximations: cos 22^ = 0.9 

cos 45 = 0.7 
cos 67| = 0.4 are mostly sufficient). 

Generally, one can be content with 8 directions. If 16 direc- 
tions are given in the original data,* it is usual to reduce these to 8 
directions. The frequencies of the intermediate directions are 
allotted by 50% to the two nearest main directions. 2 Analogous 
formulas can be given for the points of the compass. 3 

The values L of the numerical example of Table 28 4 calculated 
with Lambert's formula yield : 

C N = + 622 miles for January 1926 
Cw = + 479 miles for January 1926 

Therefore 

C N /Cw = tana' = + 622/+ 479 = 1.30 

or a' = 52, the angle between the north and the west direction 

2 Objections to this procedure are more or less justified theoretically, but in practice 
this method is a sufficient approximation. 

8 If a greater accuracy is desired or if more than 8 directions have to be considered, 
L. W. Pollak's Rechentafeln zur harmonischen Analyse (Leipzig, 1926) facilitate the 
procedure. The "Handweiser zur harmonischen Analyse," Prager Geophysikalische 
Studien, vol. II (Prague, 1928), pp. 54-59, contains special instructions for calculating 
the resultant wind direction. 

4 Data taken from Observations Made at the Royal Magnetical and Meteorological 
Observatory at Batavia (Batavia, 1930), with some handwritten corrections by H. P. 
Berlage, Jr. 



WIND 



103 



(Fig. 23). Because the azimuth of wind direction is usually 
reckoned in a clockwise direction (north through east, etc.), the 
true azimuth 

a = 52 + 270 = 322 

N 




FIG. 23. Graph for evaluating the true azimuth of the resultant wind 

VIII. 2. RESULTANT WIND DIRECTION CALCULATED FROM 
FREQUENCIES OF THE DIRECTIONS ALONE 

It often happens that only the frequencies of the single wind 
directions are published, while their average velocities are not 
known. In this case, also, it is possible to estimate the resulting 
wind direction. 

The method is based on the assumption that the average 
velocities are roughly proportional to the frequencies of the direc- 
tions. 5 The author examined the wind records for 35 months at 
Batavia, Java. The correlation between frequency of the direc- 
tions and the velocity is shown in Figure 24. If, as in this random 
sample, the assumption in question holds, then winds from the 

* For instance, if west winds are the most frequent winds at a place, they are also 
the strongest winds. 



104 



METHODS IN CLIMATOLOGY 



most frequent directions are also the strongest. Hence, in the 
first approximation, the frequencies F (first line of Table 28) can 
serve for L, the run of the wind. If we calculate with the values 
F of Table 28, we get 

a = 321 




0.5 1.0 1.5 20nvs 



FIG. 24. Correlation between the frequency of the directions and the 
average velocity of the wind, demonstrated by means of the wind records of 

35 January months at Batavia, Java. (After V. Conrad) 

* 

v. 

instead of the true value, a = 322. In this example the differ- 
ence is incidentally negligible. 6 

VIII. 3. THE RESULTANT RUN OF THE WIND 
The resultant run of the wind is given by 



ft = VCV + C w * = A/622 2 + 479 2 = 785 miles 
for Batavia, Java, January 1926. 

8 The calculation which leads to the result given above is as follows: 
The numbers of hours with the different wind directions (F) (Table 28, first line) 
are now introduced into Lambert's formulas instead of the values L. 



Therefore 



or 



and finally 



C N = (153 - 41) + 0.7( 37 + 137 - 8 - 25) + - 210.7 
C w ( 92 - 1) + 0.7(137 + 25 - 37 - 8) = + 172.9 

tana' - + 210.7/+ 172.9 - 1.2186 
a' 50 38' 
a - 320 38' 



WIND 



105 



The average resultant wind velocity v is calculated by dividing 
91 by the number of hours with wind. That means that the num- 
ber of calms is excluded. The sum of the hours with measurable 
wind velocity is in the example of Table 28 (first line) Z/ 7 = 494 
Therefore 






785 
494 



= l.S9mi/h = 0.71 m/s 7 



Frequently, Beaufort numbers are published in the year- 
books and annals; these should be converted into absolute measure, 
i.e., m/s or miles per hour, etc., for purposes of comparison. 
Therefore, Table 29 gives equivalents in meters per second, 
kilometers per hour, and miles per hour. These equivalents were 
recommended by the meeting of the International Meteorological 
Committee at Vienna, Austria, in 1927. 8 

TABLE 29. EQUIVALENTS OF THE NUMBERS OF BEAUFORT SCALE * 



Beaufort 
numbers 



m/s 



km/hour 



miles/hour 






0.25 


0.9 


0.6 


1 


1.15 


4 


2.6 


2 


2.55 


9 


6 


3 


4.30 


16 


10 


4 


6.40 


23 


14 


5 


8.65 


31 


19 


6 


11.15 


40 


25 


7 


13.85 


50 


31 


8 


16.75 


60 


38 


9 


19.90 


72 


44 


10 


23.35 


84 


52 


11 


27.10 


98 


61 


12 


>29 


>104 


>65 



* The average values are calculated from the limits in m/s, originally given by the 
International Meteorological Committee (1927) for each Beaufort degree. 



In Table 29, the Beaufort no. is equivalent to 0.25 m/s, 
owing to the fact that the record "0" (calm) includes also the very 
weak winds. 

7 The necessary tables for conversion of the English measure to the cgs-system, 
respectively centigrades, meters, etc., are given in the Smithsonian Meteorological 
Tables (Washington, 1918). 

8 See also The Meteorological Glossary (Meteorological Office, London, 1939), p. 36, 
with slightly different values. 



106 METHODS IN CLIMATOLOGY 

VIII. 4. THE STEADINESS OF THE WIND (5) 

The steadiness of the wind is the ratio of the resultant run of the 
wind (9fc) to the run of the wind (R), disregarding the direction, 
multiplied by 100. 

5 =100 1 
K 

If during the whole period considered, the wind always blew in one 
and the same direction, then 

8? = R 
S Ma * = 100% 

and we have the upper limit of S. If there is no prevailing wind 
direction and the wind shifts in a quite incidental way and there 
are a great many wind observations, the sum of all velocities will 
be zero if their direction is considered. Then 9? = 0, while R has a 
finite value (because a place with everlasting calm is impossible). 
Therefore the lower limit 

S M in = 0% 

The steadiness of the wind varies between and 1UU%, according 
to the above definition. 

In the example of Table 28, the total run of the wind, disregard- 
ing the direction (sum of second line) 

R = 1069 miles 
The resultant run of the wind (considering the direction) is: 

$ = 785 miles 

Therefore 

70 cr 



The complete result may be expressed as follows: The rainy mon- 
soon at Batavia, in January 1926, had an average net velocity of 
1.6 miles per hour, an average azimuth of 322 (W S2N) (N 38W) 
and a steadiness of 73%. These indications characterize the 
average wind conditions. The great steadiness of the monsoon 
wind in January is very significant for this phenomenon. 



CHAPTER IX 
SOME COMBINED ELEMENTS 

IX. 1. RELATIVE HUMIDITY (R.H.) 

A GRAPHICAL correlation between the monthly average rela- 
tive humidity and the average temperature, called climo- 
gram, is frequently used. Figure 25 gives an example for 
Boston, Mass. 1 The method may be valuable for purposes of 
illustration. In the present example, it might be stated that the 



72 
% 

70 

6*8 

66 

64 




20 30 40 50 60 70 F 

FIG. 25. Climogram of Boston, Mass. (42.4N, 71.1W, 125 feet) 

months August through December are more humid than the 
months March through June, at similar temperatures on the New 
England coast. 

This kind of representation is, of course, not restricted to 
relative humidity and temperature. It can be used for every pair 
of elements which are subject to an annual course and are related 
to one another. Fine examples are presented by P. Gotz, 2 who 
correlates in this way, for instance, the red and infrared content 
of radiation, turbidity factor, etc., with water- vapor pressure. 

1 G. H. Noyes, Annual Meteorological Summary with Comparative Data 1942 
(U. S. Weather Bureau. Boston, 1943). See also F. Eredia, "Sul 1 umidita relativa in 
Italia/ 1 Gerland's BeitAge zur Geophysik, vol. 33 (1931), p. 286. 

* F. W. Paul G5tz, Strahlungsklima von Arosa (Berlin, 1926). 

107 



108 METHODS IN CLIMATOLOGY 

IX. 2. EQUIVALENT TEMPERATURE () 

Equivalent temperature combines temperature with water- 
vapor pressure. For illustration: the total water- vapor content 
of one cubic meter could be condensed at constant pressure. 
By this process a certain amount of heat would be released, with 
which the dry air could be heated A/ C. 8 

If / is the observed air temperature, then the equivalent tem- 
perature is defined by the equation : 

= / + A/ 

For usual climatic temperatures, the difference 

A/ = ke 

which means, it is proportional to the water-vapor pressure, e, 
where k is the coefficient of proportionality. This coefficient is 
expressed by the following equation, given by F. Linke: 

R(607 + 0.7080(273 + 



0.239 X 13.6 X b X 1000 

where R is the gas constant (R = *29.27 gttfri* sec" 2 ), / = tempera- 
ture in degrees centigrade and b means the air pressure in milli- 
meters of mercury (Hg). This formula has been evaluated for 
different pressures and temperatures between b 770 mm Hg and 
b = 740 mm, within the limits of the temperatures 20C ( 4F) 
and 30C (86F). The calculation yields the following results: if 

/ = - 20 and b = 770 mm: k = 1.84 
/ = + 30 and b = 740 mm: k = 2.16 

In the case of simple estimations where no real exactness is re- 
quired, and at medium pressures and temperatures, it can be 
assumed that k = 2. Thus 

- / + 2e 

8 The total heat content (W) of a cubic meter of moist air is composed of the heat 
contents of its dry air and its water vapor. The equation reads: 



where c p means the specific heat of air at constant pressure, p the density, T the absolute 
temperature, / the absolute humidity in grams per cubic meter, and r the heat of vaporiza- 
tion. Dividing the total heat content by c p p, we obtain a value of temperature character. 
In rough approximation, the absolute humidity,/ (g per m 8 ) and the vapor pressure e 
are more or less equal numbers as far as usual climatic temperatures are concerned. 
(See F. Linke, Met. Zeit., 1922, pp. 268 ff.) 



SOME COMBINED ELEMENTS 109 

This formula is valid as an approximation only within the limits 
mentioned and should not be used if these limits are exceeded by 
temperature and pressure. As an example, the annual course of 
the average equivalent temperatures at Boston, Mass., is given 
(Table 30, columns 5 and 6). The other columns present the 
numerical values for calculating the equivalent temperature. 

Finally, in some cases it is desirable to figure out exact values of 
the equivalent temperature. F. Linke 4 has given another 
equation : 

= t + RH-K(t,b) 
where 

_ 1543 + 1.68< 
K(t ' b > " b - 0.377E b 

and RH = e/E. 

For evaluating the formula, the temperature / in centigrade 
degrees, the air pressure in millimeters of mercury, and the rela- 
tive humidity have to be known. The table in Appendix III is a 
good expedient, for this computation and yields directly the quan- 
tity of K(t, b}. The vertical entry is the temperature; the hori- 
zontal, the air pressure. Examples follow: 

1) / = - 18C; b = 780 mm; RH = 0.70 

from the table: K = 1.87 
e = - 18 + 0.70 X 1.87 = - 16.7C 

2) t = 0C; b = 760 mm; RH = 0.70 

from the table: K = 9.32 
= + 0.7 X 9.32 = 6.5C 

3) / = 30C; b = 740 mm; RH = 0.70 

from the table: K = 65.2 
= 30 + 0.7 X 65.2 = 75.6C 

The examples show the simple procedure of calculation by means 
of the table and the interesting effectiveness of vapor content 
upon the equivalent temperature at different air temperatures. 

The calculation is still simpler if the temperature of the wet 
bulb /' is known. Then 

= /' + K(t' 9 760) 

4 F. Linke, Meteor ologisches Taschenbuch (Leipzig, 1931), p. 287. 



no 



METHODS IN CLIMATOLOGY 



Example: if 



t' - 25.9C, = 75.7C 



This wet-bulb temperature corresponds to the temperature- 
humidity conditions of example no. 3. The two calculations of 
this example yield identical results. In the last equation, corre- 
lating /' with , one needs consider neither the actual air pressure 
nor the actual relative humidity. 

TABLE 30. AVERAGE MONTHLY EQUIVALENT TEMPERATURES 
(0 = * + 2e) AT BOSTON, MASS. 





F 
(1) 


C 

(2) 


e, mm 
(3) 


2e 

(4) 


c 

(5) 


o F 

(6) 


Tan 


27.9 


2.3 


2.6 


5.2 


2.9 


37 


Feb 


28.8 


1.8 


2.6 


5.2 


3.4 


38 


Mar 


35.6 


2.0 


3.4 


6.8 


8.8 


48 


Apr 


46.4 


8.0 


5.2 


10.4 


18.4 


65 


May 


57.1 


13.9 


7.9 


15.8 


29.7 


85 


June 


66.5 


19.2 


11.2 


22.4 


41.6 


107 


July 


71.7 


22.1 


13.6 


27.2 


49.3 


121 


Aue 


69.9 


21.1 


13.3 


26.6 


47.7 


118 


Sent 


63.2 


17.3 


10.5 


21.0 


38.3 


101 


Oct 


53.6 


12.0 


7.2 


14.4 


26.4 


80 


Nov 


42.0 


5.6 


,4.8 


9.6 


15.2 


59 


Dec 


32.5 


0.3 


3.2 


6.4 


6.7 


44 

















Column 1 : Air- temperature in F. 

Column 2: Air- temperature in C. 

Column 3: Water-vapor pressure in millimeters of mercury. 

Column 4: The values of Col. 3 doubled. 

Column 5: Equivalent temperature in C. 

Column 6: Equivalent temperature in F. 

TABLE 30a. THE EQUIVALENT TEMPERATURES COMPUTED FROM 8 = t + 2e (TABLE 
30), COMPARED WITH THE RESULTS (G') FROM THE CORRECT FORMULA (SEE p. 109).* 





c 


RH 


k(t. 760) 


KXRH 


e /o c 


ec 


<',c e) 


I 


-2.3 


0.67 


7.83 


5.2 


2.9 


2.9 


0.0 


II 


-1.8 


.64 


8.13 


5.2 


3.4 


3.4 


0.0 


III 


2.0 


.65 


10.7 


7.0 


9.0 


8.8 


+0.2 


IV 


8.0 


.65 


16.2 


10.5 


18.5 


18.4 


+0.1 


V 


13.9 


.66 


24.0 


15.8 


29.7 


29.7 


0.0 


VI 


19.2 


.67 


33.4 


22.4 


41.6 


41.6 


0.0 


VII 


22.1 


.68 


39.7 


27.0 


49.1 


49.3 


-0.2 


VIII 


21.1 


.71 


37.4 


26.6 


47.7 


47.7 


-0.0 


IX 


17.3 


.72 


29.6 


21.3 


38.6 


38.3 


+0.3 


X 


12.0 


.67 


21.1 


14.1 


26.1 


26.4 


-0.3 


XI 


5.6 


.70 


13.8 


9.7 


15.3 


15.2 


+0.1 


XII 


0.3 


.67 


9.53 


6.4 


6.7 


6.7 


0.0 



* K(t t b) is taken from Appendix HI, assuming b 
the year. 



760 mm Hg, constant through 



SOME COMBINED ELEMENTS 111 

IX. 3. DRYING POWER 

Drying power combines saturation pressure for a given actual 
or average temperature, water-vapor pressure, and wind velocity. 
A short instruction on how to calculate this combined element 5 
may suffice. The theoretical basis could be improved. The idea 
is extremely useful in the description of a climate. The definition 
reads: Drying Power is the amount of evaporation, in centimeters, 
within four hours. It is calculated by means of the following 
empirical formula: 



S 9 = 0.023 X F(v) - X (1 + 0.084 v) 

e A/ 

where 

S v = drying power at the wind velocity, v, in km/hour 

e = the observed water- vapor pressure in mm Hg 
E t = saturation pressure at the temperature, tC 
F(v) = a factor dependent on v 6 

AJ3/A/ means the variation of the saturation pressure with tem- 
perature and can be taken from any psychrometric table which 
indicates temperature in C and the saturation pressure in mm Hg. 

Example: The saturation pressure, E, is taken at one degree 
above and at one degree below the given air temperature. 

If t = 10C then from the table: E 9 * = 8.61 mm and 



OQA *u 9,84 - 8.61 

= 9.84 mm then - = ~ - - = 0.615 

A/ 11 y 

If AE/A/ is known, the computation of the formula of the dry- 
ing power offers no further difficulty or doubt. 

1 W. Knoche, " El ' Valor de Desecaci6n ' como factor climatol6gico," Revista ChileHa 
de Historia y Geografia 1919, nos. 34, 35; W. Knoche, " Der ' Austrocknungswert* [Dry- 
ing Power] als klimatischer Faktor, 11 Aus dem Archiv der Deutschcn Seewarte, vol. 48 
(1929), no. 1. 

6 The values of F(v) for different wind velocities are: 



km/hour 


F(t> 


km /hour F(v) 





1.000 


6 


1.667 


1 


1.148 


7 


1.712 


2 


1.274 


8 


1.746 


3 


1.392 


9 


1.762 


4 


1.493 


10 


1.777 


5 


1.592 


15 


1.782 



112 METHODS IN CLIMATOLOGY 

It is only necessary to add that the formula is valid only for 
sea-level. For other levels, S v has to be multiplied by b /b, 
where b = 1015 mb and b is the pressure in millibars at the place 
in question. 

IX. 4. COOLING POWER 

Cooling power is observed by the dry and by the wet kata- 
thermometer invented by Leonard Hill. This book does not deal 
with instrumental technique, but it may be noted that Hill suc- 
ceeded so far in analyzing the cooling components a to arrive at 
analytical equations which give the cooling power from values of 
temperature, wind velocity, and humidity. 

Cooling power is a bioclimatological combined element and 
is defined as follows : The cooling power is equal to the heat loss which 
a surface of a body at the temperature of the human blood (99F, 37C) 
experiences, if exposed to the free air. It is measured in milligram 
calories per square centimeter and second (mgcal/cm 2 , sec). 

There are two kinds of cooling power: 

1) Heat loss of a dry surface (H). 

2) Heat loss of a wet surface 



The equations read : 

A) for wind velocities less than 1 m/s : 

dry surface: H = (0.20 + 0.40A/*)(36.5 - /) 

wet surface: H' = H + (0.060 + 0.072^ 3 )(E - *) 4 ' 3 

H = the dry cooling power in mgcal/cm 2 , sec 
H' = the wet cooling power in mgcal/cm 2 , sec 

v = the wind velocity in meters per second 

t = the air tempersture in C 
E = the saturation pressure in millimeters of mercury 7 

e = the vapor pressure in millimeters of mercury 

B) for wind greater than 1 m/s: 

dry surface: PI = (.13 + .47Vfl)(36.5 - f) 
wet surface: H' = H + (.035 + .098z> 1/3 )( - 



In English units, i.e., when v is the velocity in feet per minute and 
/ the temperature in degrees Fahrenheit, the equations read : 

*At36.5C (97.7F). 



SOME COMBINED ELEMENTS 113 



dry surface : 

below 200 ft/min H = (.11111 + .01584^(97.7 - /) 
above 200 ft/min H = (.07222 + .0186lA/*>)(97.7 - /) 

wet surface: 

below 200 ft/min H' = (.19444 + .08 11 8^) (97. 7 - /') 
above 200 ft/min H' = (.05556 + .10505^)(97.7 - /') 

The formulas for wind velocities greater than 1 m/s or 200 
ft/min seem to hold very well ; but are perhaps not fully valid for 
the small velocities. 8 

The symbol t f of the two equations for the wet surface denotes 
the temperatures of the wet-bulb thermometer. 

Newer formulas for the dry cooling power originate from par- 
ticularly exact experiments: 9 



H = 0.99(r(7V - TV) + (.113 + .34^- 622 )(^ - T A ) 



where T K denotes the absolute temperature of the kata-thermom- 
eter, T A the temperature of the air, and <r the constant of the Stefan- 
Boltzmann-law : 



a = 8.26 X lO 
For moderate conditions, i.e., 

(T K + T\0:2 = 300a, in very rough approximation, 
the formula above can be simplified: 10 

H = (1 K - t A )(.26 + .34t/>- 622 ) 
where IK and IA mean centigrade degrees. 

8 V. Conrad, "Messung und Berechnung der Abkiihlungsgrosse," Gerland's Beitrdge 
zur Geophysik, XXI (1929), 183. For further information See D. Hargood Ash and 
Leonard Hill, "The Kata- thermometer," in Studies of Body Heat and Efficiency (London, 
1923); C. F. Brooks, "The Cooling of Man under Various Weather Conditions," MWR, 
vol. 53 (1925), pp. 423-424; E. C. Donnelly, "Human Comfort as a Basis for Classifying 
Weather," ibid., pp. 425-426; V. Conrad, Fundamentals of Physical Climatology, where 
the reader will find a diagram (p. 113) from which he can make rough estimations of dry 
cooling power for temperatures between 50F and +98F and wind velocities from 
calm to 25 mi/h. 

9 H. Lehmann, Mikroklimatische Untersuchungen der Abkuhlungsgrosse in einem 
Waldgebiet (Leipzig, 1936). See also R. G. Stone, "On the Practical Evaluation and 
Interpretation of the Cooling Power in Bioclimatology," BAMS, vol. 24 (October 1943), 
pp. 304 ff . 

10 K. BUttner, Physikalische Bioklimatologie (Leipzig, 1938), p. 98. 



114 METHODS IN CLIMATOLOGY 

For bioclimatological purposes IK has to be assumed as con- 
stant at 36.5C. It should also be emphasized, from the methodo- 
logical standpoint, that the relations of cooling power to physio- 
logical processes are very complicated. The application of cooling 
power to the problems of the heat balance n of the human body 
assumes that the temperature of the human skin is constant. This 
assumption does not hold. 

11 See C. F. Brooks and E. C. Donnelly, cited in note 7 above, and D. Brunt, in 
QJRMS, 1943. 



CHAPTER X 
CLOUDINESS 

IN THE FOREGOING DISCUSSION of variates of different kinds, the 
statistical properties of cloudiness were considered, and their 
relation to the relative sunshine duration too. (See III. 6 
and III. 10.) Especially for this element, average values have to 
be supplemented by computing frequency distributions. 1 When 
observing and discussing cloudiness, one should discriminate be- 
tween low and high clouds. 2 

X. 1. CLEAR AND CLOUDY DAYS 

In a number of national meteorological networks no detailed 
observations of the cloudy part of the sky are made. Only the 
day as a whole is characterized ' 'clear/' "cloudy," or "partly 
cloudy." In the United States, this kind of observation is usual 
at the cooperative stations, because records are made only once 
a day. 

The international definitions (Utrecht, Holland, 1874) are: 

1) Clear days are days with an average cloudiness between 0% 
and 20% of the visible sky. 

2) Cloudy days have an average cloudiness between 80% and 
100%. 

The instructions for cooperative observers of the U. S. Weather 
Bureau which differ from the international use are as follows: 
4 'The general character of the day from sunrise to sunset should be 
recorded as 'clear* when the sky averages three tenths or less ob- 
scured . . . and 'cloudy' when more than seven tenths obscured." 

A successful effort has been made to correlate the frequencies 
of clear and cloudy days with the average percentage of cloudiness. 
The records at stations where cloudiness as well as frequencies of 
clear and cloudy days are simultaneously observed, give the op- 
portunity of establishing an analytical relation between cloudiness 

1 W. KOppen, H. Meyer, " Die HSufigkeit der verschiedenen Bew6lkungsgrade als 
klimatisches Element/' Archiv D. Secwarte, XVI (1893), no. 5. 

* A. AngstrOm, "Ueber die Schatzung der BewSikung," Met. Zeit. (1919), pp. 257- 
262. 

115 



116 METHODS IN CLIMATOLOGY 

and frequency. A likely assumption is that cloudiness is propor- 
tional to the difference of the frequency of cloudy days minus that 
of clear days. If o means the average frequency of cloudy days 
and c that of clear ones, and n denotes the number of days of the 
month, season, etc., the following equations are valid for different 
regions : 

Norway C = 51 + 51 

Russian Empire C = 50 + 52 
Palermo, Sicily C = 48 + 40 



n 

o c 
n 

o c 
n 



In Spain, the same definitions are used as in the United States. 
The formula reads: 

Spain C = SO + - (o - c) 



TABLE 31. FREQUENCIES OF CLEAR (c) DAYS AND CLOUDY (o) DAYS 
AND THEIR RELATION TO CLOUDINESS (C%) 

(Example taken from the Observations at Blue Hill Meteorological Observatory) 



Month 


-C 
Days 


Cloudiness 
Obs. Calc. 

% % 


Obs. - Calc. 

% 


Jan 


4 


56 
52 
56 
53 
57 
56 
53 
52 
49 
50 
54 
54 


55 
51 
57 
55 
57 
57 
51 
51 
49 
49 
55 
55 


+ 1 
+ 1 
-1 
-2 

-1 
+2 
+ 1 

+ 1 
-1 
-1 


Feb 


2 


Mar 


5 


Apr 


4 


May 


5 


June 


5 


July 


2 


Aue 


2 


Sept 


1 


Oct 


1 


Nov 


4 


Dec 


4 



Year 3 53 53 



2(+) = 6 
Z(-) = 6 



CLOUDINESS 117 

For Blue Hill Observatory, south of Boston, Mass. (42.2N, 
71.1W, 640 ft) simultaneous observations of cloudiness and fre- 
quency (c and 0) are available. 3 

With the assumption that C = a + b (o c), one obtains the 
equation C = 47 + 2 (o c}. This equation differs from those 
previously mentioned in that no distinction is made between the 
different lengths of the months. This may be an example of 
avoiding an unfounded and exaggerated accuracy. The numbers 
in Table 31 show that the calculated values are in very good 
agreement with those observed, in spite of the simplified equation. 

X. 2. FOG 
The usual statistical characteristics are : 

1) Average number of days with fog, divided sometimes 
into different degrees of density. 

2) Probability of a foggy day ( = the number of foggy 
days divided by the number of days in the month). 

3) Average duration of fog, in hours, on a foggy day. 

The last characteristic is computed by the method of random 
samples. The procedure is identical with that of the duration of 
precipitation which was discussed earlier (III. 10). Usually more 
detailed statistics are compiled only for the oceans. Six observa- 
tions every 24 hours are made at sea, so that in this case the method 
of random samples is very successful. 

* A. McAdie, "Observations and Investigations made at Blue Hill Meteorological 
Observatory, Mass., 1915 with Summaries for Thirty Years 1886-1915," Annals of the 
Astronomical Observatory of Harvard, vol. LXXIII, pt. Ill, (Cambridge, 1916). 



i 



CHAPTER XI 

PRECIPITATION 

F THE PRECIPITATION is measured with the standard rain- 
gauge, not with a self-recording instrument, two primitive 
elements can be taken directly from the records: 

1) Number of days with measurable precipitation 1 

la) with rain 
Ib) with snow 
Ic) with hail 
Id) with sleet 

2) Total amounts of precipitation fallen during the single 
days, during a month, etc. 

Starting from these data one proceeds to the following concep- 
tions. 

XL 1. PROBABILITY (P) OF *\ DAY WITH PRECIPITATION 

This is the quotient of the number of days with precipitation 
(r) over the total number of days of the period in question 
month, season, year (n) : 

P = 100 - % 
n 

(Boston, Mass., has 125 days with precipitation in the average of 
60 years). Therefore 



P = 100 = 34.2% = 342/1000 



1 Recently H. Neuberger ("On the Measurement and Frequencies of Traces of 
Precipitation/' BAMS, vol. 25, 1944, pp. 183-188) has published interesting statistics 
on traces of precipitation measured by means of new instruments which overcome some 
difficulties of the instruments of S. P. Fergusson and others. Here, it is not the instru- 
mental side which is of interest but the fact that the investigations permit differenti- 
ating frequencies of traces and measurable precipitation. Statistics of the two phenom- 
ena separated from one another would be valuable and also, e.g., annual courses of the 
ratio of the frequencies of traces and real precipitation. Not only applied climatology 
(agricultural) would profit much. Numbers of days with traces mean a new and 
important precipitation element if the traces are observed in a well-defined way. 
In any case, this element is at least as consequential as the number of days with 
dew. 

118 



PRECIPITATION 119 

If only per cent is calculated, it suffices to take the year with 365 
days. 

XL 2. PROBABILITY THAT A DAY WITH PRECIPITATION 
Is A DAY WITH SNOWFALL 

Generally, this probability should be calculated only for the 
average of the snowy period of a climatic region (Table 32), not 

TABLE 32. PROBABILITY THAT A DAY WITH PRECIPITATION is A DAY WITH SNOW- 
FALL (Ps). BOSTON, MASS. NUMBER OF DAYS WITH PRECIPITATION AND SNOW- 
FALL, (60 AND 31 YEARS RESPECTIVELY), TAKEN FROM G. H. NOYES, 
Meteorological Summary, Comparative Data 
(U. S. Weather Bureau, Boston, 1943) 



Month 


Number of Days 
with Precipitation 


Number of Days 
with Snowfall 


PS 

% 


July 


10 








Ausr 


10 








Sept 


9 








Oct 


9 


1) 


(0) 


Nov 


10 


3 


30 


Dec 


11 


8 


73 


Jan 


12 


11 


92 


Feb 


10 


10 


100 


Mar 


12 


7 


58 


Aor 


11 


3 


27 


May 


11 


(<1) 


(0) 


June 


10 

















for the whole year. If the number of days with precipitation is 
r , and s of them are days with snowfall, the probability 

P. = 1005/r% 

XL 3. AMOUNTS OF PRECIPITATION WITHIN CERTAIN PERIODS 

The amounts of precipitation for single days are totaled for 
different sections of the year according to the purposes in question. 
There are average sums and individual amounts per week, ten 
days, month, season, year. For agricultural purposes, the amount 
of precipitation which occurs during the vegetative period (growing 
season) will be of interest (see the explanation in the chapter on 
temperature, VI. 3). The beginning and end of these periods is 
either derived from the annual course of temperature, or from bio- 
logical and phenomenological experiences; for instance, the time 
from the average date of sowing to that of the beginning of the 
flowering season. 



120 



METHODS IN CLIMATOLOGY 



In an ordinary climatography, at least the monthly and the 
annual amounts have to be given. Average amounts of precipita- 
tion should be given in millimeters; frequently better, in centi- 
meters ; or else, if possible, in hundredths of inches and otherwise, 
in tenths. This is the accuracy which corresponds to the exactness 
of the observations. 

XL 4. THE ANNUAL COURSE OF THE AMOUNTS OF PRECIPITATION 

The annual amount of precipitation may be considered as being 
evenly distributed over all days of the year. Monthly sums can 
then be calculated. The resulting course shows, however, varia- 
tions caused by the unequal length of the months. This incon- 
venience gives rise to different methods for eliminating this source 
of error. 

XL 4 a. The Reduction to Months of Equal Length 

This can be achieved in two ways. The simpler way is to re- 
duce the monthly amount to equal months of 30 days. The 
amounts for the months with 31 days are reduced by 3.2%. The 
amount for February (28.25 days) fe increased by 6.2%. Table 33 

TABLE 33. MONTHLY PRECIPITATION (MM) AT A PLACE IN CENTRAL EUROPE 
REPRESENTED BY DIFFERENT METHODS 



1 


2 


3 


4 


5 


6 


7 






Col. 2 














reduced 




Even distri- 


cart 






Average 


to equal 


Col. 2 in 


bution of 


pluviomttrique 


Relative 




monthly 


months 


thousandths of 


an annual 


relatif 


pluvio- 




precipitation 


of 30 days 


the annual 


amount of 


(Col. 4 minus 


metric 


Month 


mm 


mm 


amount 


1000 


Col. 5) 


coefficient 


Jan 


37 


36 


57 


85 


-28 


0.67 


Feb 


33 


35 


51 


77 


-26 


0.66 


Mar 


47 


45 


73 


85 


-12 


0.86 


Apr 


53 


53 


82 


82 





1.00 


May. . . . 


71 


69 


110 


85 


25 


1.29 


June. . . . 


70 


70 


108 


82 


26 


1.32 


July 


79 


76 


122 


85 


37 


1.43 


Aug.. . . 


69 


67 


106 


85 


21 


1.25 


Sept.. . . 


50 


50 


77 


82 


5 


094 


Oct 


47 


45 


73 


85 


-12 


0.86 


Nov. 


45 


45 


69 


82 


-13 


0.84 


Dec 


47 


45 


73 


85 


-12 


0.86 



Year. 



.648 



(636) 



1000 



1000 



PRECIPITATION 121 

contains in the second column the original monthly sums at a place 
in Central Europe. They are reduced to months of equal length 
of 30 days in the third column. 

The second method is to reduce the original sums (column 2) to 
a month which has the length of the twelfth of a year. That is 
30.438 days. Therefore the amounts 

for February have to be multiplied by 1.077 
for the months with 30 days, by 1.015 

for the months with 31 days, by 0.982. 

XL 4 b. Amount of Precipitation on an Average Day of the Month 

The monthly amount is divided by the number of days of the 
month in question. The advantage of this method is that the 
inequality of the months is eliminated. The great disadvantage 
however, is, that these numbers are absolutely fictitious. Thus, 
it was not of much value for F. H. Bigelow 2 to calculate " Daily 
Normals of Precipitation " for a great number of stations in the 
United States. Precipitation is always a discontinuous climato- 
logical element. Every day and everywhere there is always a 
temperature ; but there are days with precipitation and days with- 
out. An evenly interpolated distribution gives a wrong picture 
of the phenomenon. 

XI. 4 c. cart Pluviomtrique Relatif and the 
Relative Pluviometric Coefficient 

A. Angot suggested two other methods of representation, which 
though not very illuminating have been frequently used in climato- 
logical papers and books. His Ecart pluviometrique relatif may 
best be translated relative monthly deviation. In Table 33, column 
5, the numbers of an even distribution over the months are shown 
for an annual amount of 1000. In column 4, the actual distribu- 
tion per mille is calculated from column 2. In column 6, the differ- 
ences (column 4 minus column 5) are given. These differences are 
called icart pluviometrique relatif, or relative monthly deviations. 

Monthly quotients are calculated with the numbers of column 4 
over the homologous numbers in column 5. These ratios are called 

2 Frank H. Bigelow, "The Daily Normal Temperature and the Daily Normal Pre- 
cipitation of the United States/' U. S. Weather Bureau, Bulletin R (Washington, D. C., 
1904). 



122 METHODS IN CLIMATOLOGY 

relative pluviometric coefficients (column 7). 3 Both expressions are 
scientifically well founded and eliminate the inequalities of the 
months. 

All in all, the reduction to equal months of 30 days seems to be 
the best expedient. The small difference between the actual 
annual sum and the reduced sum (see last line of columns 2 and 3) 
is of no significance. 

XL 5. ABSOLUTE AND RELATIVE VARIABILITY. QUOTIENT OF 

VARIATION. METHOD OF RANDOM SAMPLES APPLIED 

TO RECORDS OF RAINFALL 4 

These three topics have been discussed in connection with 
related problems of mathematical statistics. (See III. 9 and 
III. 10). 

XI. 6. RAIN INTENSITY (I) 

The ratio between the total amount during a given period (A) 
and the number (N) of days with precipitation is called rain in- 
tensity. It is the average amount of rain of one rainy day. 



~N 

In a 50-60 years' average for August, the amount of rain at Boston, 
Mass., is A = 3.62 inches, and the number of rainfall days is 
N = 10. Therefore the rain intensity of August at Boston is 
/ = 3.62/10 = 0.36 inches = 9 mm per rainy day. The corre- 
sponding data for Cherrapunji in the Khasi Hills in British India 
(25.3N. 91.8E), in June, the month of maximum, are: A = 2632 
mm; N = 24.8; / = 106mm per rainy day. Cherrapunji is 
located in one of the rainiest regions of the world. There rain 
intensity in the wettest month is twelve times greater than at 
Boston. 

Rain intensity is a very significant characteristic of a climate. 

1 Dr. C. F. Brooks kindly draws my attention to two papers of B. C. Wallis, "The 
Rainfall of the Northeastern United States" and "The Distribution of the Rainfall in 
the Eastern United States," MWR, vol. 43 (1915), pp. 11-14 and 14-24, where the 
pluviometric coefficient is applied to the rainfall in the eastern United States. 

4 Following a British custom, rainfall, being the shorter expression, is often used 
instead of precipitation. 



PRECIPITATION 123 

XL 7. WET AND DRY SPELLS 

The average and extreme lengths of wet and dry spells are of 
decisive influence upon organisms. A wet spell is defined as a 
period of consecutive days with at least 0.01, 0.05, etc., inches, or 
0.1, 0.2, 0.5, etc., millimeters of rain. The thresholds on which the 
definition is founded have to be expressly indicated because no 
general agreement exists in this respect. 

It is assumed that about ten years' records are at hand. This is 
sufficient. A much shorter period should not be chosen. If the 
ten years are available, the length of the spells (wet or dry periods) 
should be excerpted from the monthly records. An isolated rainy 
day means a spell of one day for statistical reasons. Then 

a) the average length of a wet (dry) spell 

b) the average length of the longest wet (dry) spells 

c) the extreme lengths 

are computed for each month. If a spell overlaps from one month 
to the next, or if its length is greater than one month, the rules for 
hot and cold spells govern (see VI. 4). 

A dry spell means a period of at least five consecutive rainless 
days. It goes without saying that the establishment of a minimum 
limit is necessary; a shorter period is unlikely to damage the 
vegetation in any way. "Dry periods 1 ' longer than five days are 
not considered as being interrupted by a rain less than 1mm 
0.04 inch). 

On the coast of the middle Adriatic Sea, dry spells up to the 
extreme of 81 days are recorded in summer. The longest rainy 
period observed in the eastern Alps within 40 years had a duration 
of 16 days (in September). These two examples serve to illus- 
trate dry and wet spells. 

The definitions mentioned are by Julius Hann, slightly modified 
by the author. It is regrettable that there is no international 
definition of wet and dry spells. 

In the British Meteorological Service, other definitions are 
used : 6 

Drought is dryness due to lack of rain. Certain definitions have been 
adopted in order to obtain comparable statistical information on the 
subject of droughts. An absolute drought is a period of at least 15 con- 

1 The Meteorological Glossary (3rd ed., Meteor. Off. London, 1939). See pp. 68, 161, 
212. 



124 METHODS IN CLIMATOLOGY 

secutive days, to none of which is credited 0.01 in. of rain or more. A 
partial drought is a period of at least 29 consecutive days, the mean daily 
rainfall of which does not exceed 0.01 in.; (the word mean is to be em- 
phasized). A dry spell is a period of at least 15 consecutive days to none 
of which is credited 0.04 in. rain or more. 

Rain spell is a period of at least 15 consecutive days to each of which 
is credited 0.01 in. rain or more; thus the definition of the term rain spell 
is analogous to that of the term " absolute drought. 

Wet spell. Analogous to the term dry spell, the wet spell is a period 
of at least 15 consecutive days to each of which is credited 0.04 in. of 
rain or more/' 

These definitions are surely well fitted to the climate of the British 
Isles. The definitions by Hann-Conrad have perhaps more gen- 
eral applicability and are less complicated. 

The American definition of drought reads: " period of 20 (or 30) 
consecutive days or more without 0.25 inch of precipitation in 
24 hours during the season, March to September, inclusive." 

XL 8. CHARACTERIZATION OF THE HYDROMETEORIC 
TYPE OF A PERIOD 

Frequently it is necessary to give* a general characterization of a 
week, month, year, etc., as far as rainfall is concerned. The ques- 
tion as to whether a month is to be called dry, for instance, or 
extremely dry, wet or extremely wet, etc., in a given climate, is 
answered in III. 3. By means of the standard deviation <r, the 
problem is solved in a well-defined and reproducible way. 

XL 9. SNOW 

Table 34 shows a form representing snow conditions which has 
proved to be advantageous. The form serves for a single year as 
well as for a series of years. In the latter case, averages replace the 
individual values. If possible, extreme values taken from the 
period in question should be inserted; e.g., between columns 6 and 
7, the earliest and the latest date of both the beginning of the first 
and the ending of the last snow-cover. This makes 4 columns 
more, with increased inconvenience for printing. A supplemen- 
tary table of extremes is often more convenient. 

In Column 7 is shown the number of days on which a snow- 
cover actually exists. 



PRECIPITATION 



125 



In contradistinction to this characteristic, the season of snow- 
cover has to be indicated in Column 8. The season means the 
number of days between the date of the beginning of the first 
snow-cover and that of the end of the last one. Within the "sea- 



TABLE 34. FORM FOR TABULATING SNOW CONDITIONS FOR A NUMBER OF STATIONS 

Period to ( years) 

[Remarks regarding observation-hour and methods] 



10 



11 



12 











Date of the 










0) 


^ 












S 








5 


3 










M 











s 


* 


M 










* 







1 




i 


1 


J 


I 




CU 


-o 


JS 

M 




CO 


CO 

-S 


S 





c 
co 

^ 




i 

CO 


1 


I 


3 
5 
1 

IS 


1 
is 


Beginning 


0) 

-o 
W 


CO 

1 

"o 


8 



CO 


c 

CO 

JC 


CO 

1 


If 

*T3 O 


O 

a 

0) 


*0 


0) 


a 


a 






Ui 


O 


t 


u 


C 


3 U 




TD 


S 


2 




J 


c 


o 


-Q 


^O N ' 


s.s 


oJ 


I 


i 




Snow cover 


S 

3 


S 


rt 

Q 


3 


II 

C/J co 


X _- 

II 



















































son" the snow-cover can melt away and be formed anew. The 
number of days with an actual snow-cover is called duration of snow- 
cover, and is always smaller than, or at the most, equal to the 
season. The greater the elevation and the geographical latitude, 
the smaller the difference between season and duration. For 
example, in the eastern Alps at one place at 4000 feet altitude the 
difference is 20 days, at another place at 700 feet it is 52 days. 

The data of column 11 can be given only if the freshly fallen 
snow is regularly measured. These numbers are totaled for the 
month or the year. 

The records of the cooperative stations of the U. S. Weather 
Bureau enable one to compile statistics of the lengths of the single 



126 METHODS IN CLIMATOLOGY 

periods of the snow-cover. This is important especially for regions 
where the snow surface offers easy transportation, e.g., logging, 
skiing, etc. 

The variability of the duration as well as of the season is of im- 
portance and of interest. It decreases with elevation under other- 
wise equal conditions. Average values of duration and season 
have to be reduced to a standard period by the method of differ- 
ences. Values derived from different periods are not comparable. 

If Z means the duration in days and 5 the annual amount of 
melted snow in millimeters or inches, then the ratio Q = Z/S 
yields the duration value (Q) in days for 1mm (inch) of snow. 
This expression can be useful in some investigations. 

The percentage of the total amount of precipitation which falls 
in the form of snow should be indicated in every regional descrip- 
tion of snow conditions. 

In more detailed studies into the depth of snow on the ground, 
average month or half-month values are calculated. The statistics 
of longer series of years yield probabilities of certain average 
depths of snow in certain weeks, half months, etc. 6 

In mountainous countries, the depth of snow on the ground is, 
of course, dependent to a high degree upon altitude. Therefore, 
also, cartographic representations oT depth, duration, season, etc., 
can be based only on the principle of isanomals. Further details 
are beyond the scope of the present work. 7 

8 For a good pattern see R. G. Stone, <4 The Distribution of the Average Depth of 
Snow on Ground in New York and New England: Method of study," Transactions of the 
Am. Geophysical Union, 19th Annual Meeting (1938), pp. 486-492, and the second part, 
"Curves of Average Depths and Variability," ibid. (1940), pp. 672-692. 

7 For further discussion see " Mitteilungen (iber die Schneedecken-Verhciltnisse in 
den Ostalpen, 11 in Gerland's Beitrage zur Geophysik, as follows: (1) V. Conrad and M. 
Winkler, " Beitrage z. Kenntnis d. Schneedecken-Verhaltnisse," vol. 34 (1931), p. 473; 
(2) V. Conrad, "Die Schneedeckenzeit, ihr Anfangs-und Enddatum in den Ostalpen, " 
vol. 43 (1934) p. 225; (3) V, Conrad, "Der Anteil des Schnees am Gesamtniederschlag 
und seine Beziehungen zu den Eiszeiten," vol. 45 (1935), p. 225; (4) F. Steinhauser, 
"Ueber den Schneeanteil am Gesamtniederschlag im Hochgebirge der Ostalpen," vol. 46 
(1936), p. 405; (5) E. Ekhart, "Die Andauer der Schneedecke nach Stufenwerten der 
SchneehShe," vol. 50 (1937), p. 184; (6) V. Conrad and O. Kubitschek, "Die Veriinder- 
lichkeit und Machtigkeit der Schneedecke in verschiedenen Seehohen," vol. 51 (1937), 
p. 100. 

Additional Remark: According to an investigation in a rather extensive material, 
the variability of the depth of snow on the ground is so great that a reduction to a given 
period is not successful. Therefore, the discussion only of synchronous series gives 
reliable results. 



PRECIPITATION 127 

XI. 10. INDEX OF ARIDITY 

This concept represents an element combined of hydrometeoric 
elements and temperature elements. First, E. de Martonne's 
index may be mentioned (in a modified form) : 

7 = r+io 

where 

n = the average number of rainy days within the period in 

question, 

r = the average rainfall in the same period, 
t = the average temperature. 

The three variables are related to a certain month, a season, 
or a year. 

Another index of aridity has been established by W. Gorczyn- 
ski. 8 His aridity coefficient (AC) considers the influence of lati- 
tude, of the range of temperature, and E. Gherzi's measure of 
variation of precipitation (M - m)/p. (See III. 9.) The funda- 
mentals of the AC are, however, not yet so well investigated that 
its routine use can be recommended. 

Generally, such indices may be applied only with great caution. 
Often the results are satisfactory in one region and show contradic- 
tions in another. Furthermore, the boundary conditions are 
sometimes not fully considered, so that further investigations 
appear desirable. 

The numerical indices calculated from these formulae yield 
relative values fitted only for comparison of the conditions at 
different stations. Other similar attempts (viz., continentality, 
border lines of climatic types) are discussed in part XVII. 1 and 2. 

XI. 11. RAIN-HISTOGRAM 

Two methods of graphical representation of the annual course 
of rainfall are usual : 

1) the histogram 

2) the smooth curve. 

Both kinds are known from the representation of the frequency 
distribution (see Fig. 4). The histogram originates from the idea 

8 The Aridity Coefficient and its Application to California (The Scripps Institution of 
Oceanography, La Jolla, Calif., 1939). 



128 METHODS IN CLIMATOLOGY 

already mentioned that precipitation is a discontinuous element. 
Consequently, this representation is logical. The disadvantage of 
the method is the difficulty of drawing, and the inaccuracy result- 
ing therefrom. On the other hand, nobody can forget or overlook 
the fact that rainfall is a discontinuous element; altogether the 
smooth curve is preferable. It is more easily drawn and better 
comparable with other elements. 



PART III 
METHODS OF SPATIAL COMPARISON 

CHAPTER XII 

COMPARISON OF OBSERVATIONAL SERIES OF DIFFERENT 

PLACES. GEOGRAPHICAL DISTRIBUTION OF 

CLIMATOLOGICAL ELEMENTS 

XII. 1. UNIFORM TENDENCY OF THE VARIATIONS OF AVERAGE 

WEATHER OVER LARGE REGIONS. CONSEQUENCES DRAWN 

FROM THIS TENDENCY FOR COMPARING SERIES OF 

ELEMENTS OBSERVED SIMULTANEOUSLY AT 

DIFFERENT PLACES 

IN THE BEGINNING of the first part of this work, some remarks 
were made about different kinds of observations of climato- 
logical elements, as far as they are of methodological interest. 
It was there demonstrated (I. 3. a.) that a change of instrument, 
even if it means an important improvement, can cause a break in a 
climatological series. 

Another very trivial example may be added here. At a " secu- 
lar " station, the rain gauge happened to leak after having been 
used for some years. The hole in the bottom was very tiny in the 
beginning, so small that it closed by itself when dust and dirt were 
caught with the rain. Then for a time the gauge was watertight. 
It began to leak again when the dirt was washed away by heavier 
rain. The acids contained in the rainwater widened the hole in 
the course of time till the vessel leaked so much that the observer 
became aware of the defect. Then the gauge was repaired or 
exchanged, and the cycle began anew. The lifetime of the gauge 
was determined by workmanship and material. 

This is the story of one of the many thousands of rain gauges 
in the world. And what does it mean for climatological methods? 

As long as the rain gauge is intact, the annual amounts may 
fluctuate around a certain mean. This decreases when the gauge 
leaks, increases a little when the hole is nearly closed by chance, 
decreases again, and so on. The mean happens to reach its 

129 



130 



METHODS IN CLIMATOLOGY 



former value in the period after a new gauge is installed. Thus, 
a false cycle of a semi-periodical variation of rainfall comes into 
existence. This simple and trivial story is obviously of great 
importance to the student of periodicities in long climatological 
series. No doubt in this regard many a scientist has been misled. 
A series of observations such as the one described, showing 
variations caused by unnatural influences, is called inhomogeneous. 
It is too easy to find manifold causes of similar inhomogeneities to 
make further explanation necessary here (growing trees, new build- 
ings close to the shelter, another observer, etc.). Accordingly, the 
following definition is offered: A numerical series representing the 
variations of a climatological element is called "homogeneous" if the 
variations are caused by and only by variations of weather and climate. 




-20 



FIG. 26. Variations of January temperature at Moscow in the years 1860-1870. 
(Deviations from the normal, F) 

Obviously, it must be the first step of the climatologist to 
examine whether or not the series is homogeneous, in order to 
avoid inferences which are strictly false. This examination en- 
counters a great difficulty. There are real climatic variations from 
year to year. The example of Figure 26 shows the variations of 
average January temperatures at Moscow in the years 1860 to 
1870. We know that the observations are incontestable. If we 
did not know this fact, we should not be able to indicate whether 
the increase of temperature, e.g., from 1862 to 1863, was caused 
by nature or otherwise. Therefore we have to conclude that it 



COMPARISON OF OBSERVATIONAL SERIES 131 

is not possible to discriminate between natural and artificial varia- 
tions of an element if the observations of only one place are 
available. 

In other words, it is generally impossible to decide whether or 
not a series of observations is absolutely homogeneous. On the 
other hand, it is clear that the variations of weather and climate 
are not restricted to a single place, with the exception of a cloud- 
burst over one spot, or kindred phenomena. This principle be- 
comes better exemplified, of course, if averages over a month, a 
season, etc., are considered. In longer periods, local extreme devia- 
tions are compensated and the trend of the variation is maintained 
over large regions. 1 

First, temperature may be taken as an example. New York 
City and New Haven, Connecticut, are 80 miles apart. The 
January temperatures of the ten years reproduced in Table 35 

TABLE 35. THE QUASI-CONSTANCY OF THE DIFFERENCES BETWEEN JANUARY 
TEMPERATURES AT Two PLACES, 80 MILES APART 

New York City (40.7N, 74.0W, 314 ft) (7) 
New Haven, Conn. (41.3N, 72.9W, 106 ft) (H) 

A 7 ~ Deviations Y A# = Deviations H 



(1) 

Year 


(2) 


(3) 
H , F 


(4) 
(Y -f/)F 


(5) 
AF, F 


(6) 
A#,F 


(7) 


1911 


34.5 


32 1 


+ 2 4 


+3 5 


4-2 9 


4-0 7 


1912 


23.2 


21.2 


2.0 


7 8 


80 


T u.i 
+0 3 


1913 


. . 39.7 


38 2 


1 5 


+8 7 


4-9 


2 


1914 


31.1 


29 3 


1 8 


+0 1 


4-0 1 


4-0 1 


1915 


33.8 


330 


8 


+2 8 


4-3 8 


1 u.i 
Q 


1916 


35.1 


330 


2 1 


+4 1 


+3 8 


4-0 4 


1917.. . 


32.1 


30 6 


1 5 


+ 1 1 


+ 1 4 


2 


1918 


21.3 


19.8 


1.5 


-97 


9 4 


2 


1919 


34.9 


33 8 


1 i 


+3 9 


-4-4 6 


6 


1920 


23.8 


21.4 


2.4 


-7.2 


-7.8 


4-07 



Mean 31.0 29.2 1.7 



24.2 25.6 2.2 

24.7 25.2 2.1 

4.89 5.08 =t0.43 



show great variations. The temperature in New York varies by 
chance by the same amount as in New Haven, i.e., 18.4 F. The 
differences vary by 1.6 F. 

1 On the other hand, it is very unlikely that the same kind of artificial influences occur 
synchronously at two different places. 



132 METHODS IN CLIMATOLOGY 

The mean deviations (average variabilities) are (column 5, 6): 



AF = 4.9; Atf = 5.1 

where F means New York and H means New Haven. 
The average variability of the differences is, however: 

A(F- H) = 0.4 

Thus, the differences of the temperatures can be called quasi- 
constant, because their variations are the twelfth part of those of 
the actual temperatures. Similar conditions are clearly demon- 
strated in the example shown in Table 36A, 2 for precipitation, even 

TABLE 36A. THE QUASI-CONSTANCY OF THE RATIOS BETWEEN RAINFALL IN JULY 
AT TWO PLACES, 11 MILES APART 

Boston, U. S. Weather Bureau (42.3N, 71.1E, 124 ft) (B) 
Waltham, Cambridge Water Works (20 ft) (W) 



(1) 

Year 


(2) 
Boston (B) ' 
in. 


(3) 
Waltham (W} 
in. 


(4) 
I 
B/W 


(5)* 
AS 


(6)* 
AW 


(7)* 


1911 


4.65 


4.83 


0.96 


+0.69 


+0.73 


-0.02 


1912. .. 


5.16 


4.38 


1.18 


> +1.20 


+0.28 


+0.20 


1913. . 


. .2.69 


3.29 


0.82 


% -1.27 


81 


16 


1914. . 


2.64 


2.46 


1.07 


-1.32 


-1.64 


+0.09 


1915. . . 


8.85 


9.87 


1.07 


+4.89 


+ 5.77 


-0.08 


1916 


5.67 


5.31 


0.95 


+ 1.71 


+ 1.21 


+0.09 


1917. . 


1.10 


0.93 


1.18 


-2.86 


3.17 


+0.20 


1918... 


2.64 


2.99 


0.88 


-1.32 


-1.11 


-0.10 


1919 


4.63 


5.21 


0.89 


+0.67 


+ 1.11 


-0.09 


1920... 


....1.56 


1.78 


0.88 


-2.40 


-2.32 


-0.10 



.3.96 4.10 0.98 

9.16 9.10 +0.58 

2(-) 9.17 9.05 -0.55 

v = db 1.83 1.82 0.11 

v r (B t W) = db 45%f v r (B/W) - db 11% 

* Deviations from the means of the period 1911 to 1920. 

f v r (B t W) denotes the average relative variability of the amounts at B and W. 

if here for this element a much smaller distance between the two 
places has been chosen. If precipitation is concerned, it is not 

'Temperature data are taken from H. Helm Clayton, World Weather Records 
(Smithsonian Miscellaneous Collections, vols. 79, 90); precipitation data from X. H. 
Goodnough, "Rainfall in New England," (Journal of the New England Water Works 
Association, Boston, June 1930.) 



COMPARISON OF OBSERVATIONAL SERIES 133 

the differences that count, but the ratios between the amounts at 
the two places. The explanation will be given later on. 

Here, the ratios are much less variable than the rainfall itself, 
as is indicated by the mean deviations (v) at the bottom of columns 
5, 6, 7. In reality, this behavior of the variability of the ratios is 
not yet decisive. In this example, it is a question of the amounts, 
which means that relative, not absolute, variabilities (v r ) have to 
be considered. (See III. 9.) Therefore, the absolute variabilities 
must be divided by the respective arithmetic means (ni). These 
relative variabilities (v r ) read : 

v r (B, W) = 45% 
and 

Vr(B/W) = 11% 

So it is clear that in the example the ratios between the amounts 
can be assumed as quasi-constant, in comparison with the actual 
rainfall amounts. 

The quasi-constancy of the ratio B/W can also be shown by 
another very common representation. In Table 36n, the amounts 

TABLE 36e. FLUCTUATIONS OF RAINFALL AMOUNTS AS PERCENTAGE OF THE 
AVERAGE VALUES AS ILLUSTRATIVE (AFTER L. W. POLLAK) OF THE 

QUASI-CONSTANCY OF B/W 



Year 


Percentage of average 
Boston ^ 


1911-1920 

IValtham 


Ratio 
B/W 


B 

A 
W 
units of second 
decimal place 


1911 


117 


118 


0.99 


- 3 


1912 


130 


107 


1.21 


19 


1913 


68 


80 


0.85* 


-17 


1914 


67 


60 


1.12 


10 


1915 


223 


241 


0.93 


- 9 


1916 


143 


130 


1.10 


8 


1917 


28* 


23* 


122 


20 


1918 


67 


73 


0.92 


-10 


1919 


117 


127 


0.92 


-10 


1920 


39 


43 


0.91 


-11 













\1.02 
ifi 3.96 in. 4.10 in. 0.97 \ 11.7 



of rain in B and W are given as percentage of average, 1911 to 
1920. The amounts of rainfall at Boston vary from 28% to 
223% or by 195%, and those for Waltham by 218%, while the 
ratios of B/W vary by 37% only. 



134 METHODS IN CLIMATOLOGY 

These examples show that the variations of average weather 
particularly have identical tendencies over rather large regions. 
If the temperature drops at a place a it may drop too at the place b 
(not too far distant from a), and the same is true for rainfall, 
cloudiness, etc. In other words, there is no reason why variations 
of differences or ratios should be systematically influenced by the 
average weather events. 

XII. 2. CRITERIA OF RELATIVE HOMOGENEITY 

As previously stated, a series of numbers representing the 
variations of a climatic element is called homogeneous if the varia- 
tions are caused by and only by climatic influences. Now we have 
shown that these cannot give rise to systematic variations of " differ- 
ences" or " quotients." 

The two facts together yield an exact definition of relative 
homogeneity, as follows : A dimatological series is relatively homoge- 
geneous with respect to a synchronous series at another place if the 
differences (ratios} of the pairs of the homologous averages represent a 
series of random numbers which satisfies the law of errors. 

To determine whether or not two series are relatively homoge- 
neous is now a problem of statisticaKmathematics. I want to give 
two criteria, 3 those of Helmert and Abbe. 

XII. 2. a. Helmert' s Criterion 

The criterion of F. R. Helmert, well known in geodesy, is ap- 
plicable to the present problem and offers a first estimation. If 
two consecutive deviations of the series have the same sign, we 
speak of a sequence. The number of sequences (4- + or ) 
is called S. A change of sign of two consecutive elements is called 

change (H or h). The number of changes is called C. 

Now it may be assumed that the elements of a series are not 
systematically influenced but represent random numbers. Devia- 
tions are calculated. The consecutive signs yield 5 sequences and 
C changes. Then Helmert's criterion indicates that 

5 - C = 
with a standard deviation 



db AW - i 

V. Conrad, " Homogenitatsbestimmung meteorologischer Beobachtungsreihen," 
Met. Zeit., 1925, p. 482. 



COMPARISON OF OBSERVATIONAL SERIES 135 

where n equals the number of elements. In other words: in the 
absence of a systematic influence 



- Vn - 1 S (S - C) S + A/n - 1 
In the example of Table 35, the number 

n = 10 

therefore Vn 1=3 and 

- 3 ^ (S - C) s + 3 

Counting sequences and changes in column 7, we get : 

S- C = 3 - 6 = - 3 

Thus, Helmert's criterion shows that the series of differences may 
not be systematically influenced. This means that the two series 
of January temperatures for the years 1911 to 1920, one taken at 
New York, the other at New Haven, are relatively homogeneous. 4 

XII. 2. b. Abbe's Criterion 

Another criterion, that of Carl Abbe, which originated from 
physical research, has been suggested by the author. 6 

There may be a series of deviations from an arithmetical mean : 

From this series, two other series are derived : 

tn 
1 \ A *] 1 \ *] ^ \ j-7 2 I .^ 2 T 



i-l 

2) B = (d, - dtf + (d* - d*Y 

+ (<*<- d i+l y (d _, - <*)' + (d n - dtf 

= L" (di - d i+ tf 

=-! 

Series B is considered as cyclic, so that the last term is formed as 
the difference of the last and the first deviation. By this proce- 

4 Here a series of only ten elements is given, for the sake of brevity. Generally, 
the numbers of elements should be rather greater to permit application of probability 
criteria. See F. R. Helmert, Die Ausgleichsrechnung . . . (2nd ed., Leipzig, 1907), 
p. 333. 

V. Conrad, Met. Zcit. t 1925. 



136 METHODS IN CLIMATOLOGY 

dure, the numbers of items of the two series A and B are equal. 
Series B reads: 

t'n =n tn t n 

B = y (d d - ") ^ = y (d ) ^ I y^ f d ' i } ^ - 2 T^ d' d 
t-i *-! t-i t-i 

Now 

53 (^) 2 53 (^-i-i) 2 
t-i ==i 

as identical values. Therefore 

or 

B = 24 2 53 ^fiJi-M = 24 F 
The most probable value of 

7-^ | j | j r\ 



if the deviations are random numbers and n is a great number. 
Under these conditions one can imagine that it is always possible 
to combine any product db ;+i with another term of equal abso- 
lute magnitude and opposite sign. Consequently, the series F 
determines whether or not there is a systematic influence. If there 
is none, or in other words, if the deviations are random numbers, 
F = in so far as n is a large number. 

In concrete cases, F only approximates the value of zero with 
increasing n so that 

2A l 



(Appendix II contains the values of 1/V# for n = 1 to 119, so that 
the limits can be easily determined.) This is Abbe's criterion, 
which is fulfilled if the series in question is not systematically 
influenced. 

When we have the problem which arises from the series of 
differences or quotients, respectively, the application of Abbe's 
criterion is most valuable because it considers the sequence of the 
signs as well as the quantity of the single deviations. 6 Tables 37 

6 A third criterion regarding groups of equal signs is given by V. Conrad, "Die 
klimatologischen Elemente," in K&ppen-Geiger, Handbuch der Klimatologic, vol. IB, 
p. 114. 



COMPARISON OF OBSERVATIONAL SERIES 



137 



TABLE 37. ABBE'S CRITERION APPLIED TO A SERIES OF JANUARY TEMPERATURES AT 

NEW YORK (Y) AND NEW HAVEN (H) 

(See Table 35, Column 7) 



Year 



- A(F - 



\dt - 



1911 


+0.7 


0.4 


.49 


.16 


1912 


+0.3 


0.5 


.09 


.25 


1913 


-0.2 


0.3 


.04 


.09 


1914 


+0.1 


1.0 


.01 


1.00 


1915 


-0.9 


1.3 


.81 


1.69 


1916 


+0.4 


0.6 


.16 


.36 


1917 


-0.2 


0.0 


.04 


.00 


1918 


-0.2 


0.4 


.04 


.16 


1919 


-0.6 


1.3 


.36 


1.69 


1920 


+0.7 


0.0 


.49 


.00 



2.53 



B 5.40 



TABLE 38. ABBE'S CRITERION APPLIED TO THE RAINFALL SERIES OF TABLE 36A* 



(1) 

Year 


(2) 


(3) 
\dt -d,+i| 


(4) 


(5) 
(di <f'+i) J 


1911 


-0.01 


.22 


.0001 


.0484 


1912 


+.21 


.39 


.0441 


.1521 


1913 


-.18 


.28 


.0324 


.0784 


1914 


+.10 


.17 


.0100 


.0289 


1915 


-.07 


.05 


.0049 


.0025 


1916 


- .02 


.23 


.0004 


.0529 


1917 


+.21 


.30 


.0441 


.0900 


1918 


- .09 


.01 


.0081 


.0001 


1919 


-.08 


.01 


.0064 


.0001 


1920 


-.09 


.08 


.0081 


.0064 








A = .1586 


B - .4598 



* The series of deviations in column 2 of Table 38 differs slightly from column 7 of 
Table 36A. The result is valid for both series because that of Table 36A gives: 

2A/B = 0.77 

and 38 give examples of the application of Abbe's criterion. Here 
the series of differences (column 7 of Table 35) and the series of 
quotients (column 7 of Table 36A) are examined for relative 
homogeneity. 

From Table 37 may be calculated: 



2A 
B 



2 X 2.53 
5.40 



= 0.94 



As n = 10, 1/Vn = db 0.32 (See Appendix II). Therefore 



0.68 s r = 1-32 

> 



138 METHODS IN CLIMATOLOGY 

if no systematic influence exists. The value 2A/B lies indeed 
between these limits. Therefore it is probable that the two series 
are relatively homogeneous. 

The criteria of Abbe and Helmert arrive at the same result. 
From Table 38 we derive: 

2A _ 2 X 0.1586 
B " 0.4598 ~ U<b 
and because 



0.68 S-Sl.32 

> 

Abbe's criterion is fulfilled and there is no reason to assume a 
systematic influence upon the A(3/WO series. 7 

Counting the sequences (5) and the changes (C) of column 2 
(Table 38) it follows that: S = 3, C = 6, and 5 - C = - 3. 

As A/n 1=3, Helmert's criterion is fulfilled if 

- 3 S (S - C) s + 3 

and again the two criteria agree with one another. 

If the relative homogeneity between the series of an element at 
two places a and b is given, and a third station c, suitably located, 
is available, a conclusion can be drawn regarding the absolute 
homogeneity of one of the series. If the pairs of series (a, 6), 
(b, c), and (c, a) are relatively homogeneous, it is rather probable 
that each of the three series is absolutely homogeneous. Thus a 
sort of climatological triangulation can be accomplished. The 
larger the number of places included, the greater is the likelihood 
that the conclusions regarding absolute homogeneity will be true. 

The chief trouble with all these investigations is the scarcity 
of stations. The meshes of the climatological networks are gen- 
erally too wide, so that the application of the higher and exact 
methods meets with difficulties. Nevertheless, the examination of 
a pair of records for relative homogeneity is fundamentally im- 
portant if the climate of a region is to be described accurately. 
Only by this examination can the natural climatic changes be 
separated from the unnatural by means of an exact analysis. 

One should never forget that a leaking gauge can counterfeit 
the finest periodicity. 

7 It may be emphasized that Abbe's criterion, in the form discussed here, cannot be 
applied to smoothed series. See V. Conrad and O. Schreier, "Die Anwendung des 
Abbe'schen Kriteriums auf geophysikalische Beobachtungs Reihen," Gerland's Beitrdge 
zur Geophysik, XVII (1927), 372. 



COMPARISON OF OBSERVATIONAL SERIES 



139 



XI L 3. REDUCTION OF CLIMATOLOGICAL AVERAGES 
TO A CERTAIN PERIOD 

XII. 3 . a . Method of Differences 

Hitherto it has been assumed that the observations at the 
various places are operated synchronously. Then the comparable- 
ness is guaranteed as long as the series are mutually relatively 
homogeneous. 

TABLE 39. TEMPERATURE CONDITIONS OF DIFFERENT PERIODS 

AT ST. PAUL, MINNESOTA (45.0N, 93.0W, 

837 FT.) INSANITARY 



(1) 

Year 


(2) 
January 
F 


(3) 
Year 


(4) 

January 
op 


(5) 
Col. 4 - Col. 2 


1883 


0.9 


1898 


22.8 


+ 21.9 


1884 


7.5 


1899 


13.8 


6.3 


1885 


.... 4.3 


1900 


21.2 


16.9 


1886 


... 3.7 


1901 


16.3 


12.6 


1887 


0.7 


1902 


18.7 


18.0 


1888 


-1.1 


1903 


15.3 


16.4 


Mean 


2.7 




18.0 


15.3 













Series of different periods, however, are not comparable with 
one another. A short example is given in Table 39. 8 If one did 
not know that the two series (columns 2 and 4) originated in the 
same place, one might suppose that the series in column 4 repre- 
sented the temperatures at a place 8 latitude farther south. In 
addition, the July temperatures are higher in the period of cold 
winters, and not only are the actual temperatures very different 
in the different periods, but also the annual ranges. 

Another example (Table 40) gives the amounts of rainfall 
during different periods at Maiden Island in the South Pacific. 

TABLE 40. RAINFALL AT MALDEN ISLAND (4S, 155W) IN DIFFERENT PERIODS 



Period 



Average amount 
inches 



Years 1890 to 1899. 
Years 1911 to 1920. 
Januarys 1890 to 1899. 
Januarys 1911 to 1920. 



.16.30 
.41.97 
. 0.55 

. 5.55 



Data drawn from R. DeC. Ward and C. F. Brooks, "The Climates of North 
America," Kttppen-Geiger, Handbwh der Klimatologic, pt. J. (1936). 



140 METHODS IN CLIMATOLOGY 

There, to the best of the author's knowledge, the greatest known 
contrasts occur. 

From these examples, it is clear that the changes in the climatic 
elements with time are so effective that a comparison between 
average values of temperature, rain, cloudiness, etc., at different 
places can lead to a reasonable conclusion only if identical periods 
are considered. In view of the scarcity of climatological data, 
however, the restriction of comparisons to a certain period would 
seriously impede climatographic work. The solution of this funda- 
mental problem is found in the quasi-constancy of differences 
(quotients) of synchronous averages of different climatic elements, 
which has been discussed earlier. We take two more examples 
from Table 35: 

January New York ( Y) New Haven (H) (Y + H)/2 ( Y - H) 

1916 35.TF 33.0F 34.1 2.1F 

1918 21.3 19.8 20.5 1.5 

Diff 13.8 13.2 13.6 0.6 

This means that the January temperature changed by 13.6 F 
from 1916 to 1918, but the difference between Y and H changed by 
only 0.6 F. This behavior has been called "quasi-constancy" of 
the differences. 

Another example from Table 36A, shows the following con- 
ditions: 

RAINFALL, IN INCHES 

Boston Waltham Ratio B/W 

1915 July 8.85 9.87 0.90 

1917 July 1.10 0.93 1.18 

In other words, while the rainfall was 9 times greater in the 
wettest July (1915) than in the driest July (1917) the ratio of the 
rainfall in Boston over that in Waltham changed by only 24%. 

Thus the quasi-constancy of differences (quotients) is evident, 
both by means of a general geophysical conception and by prac- 
tical examples. 

A climatological station S 9 has been operated N years, where 
N is a relatively large number, perhaps greater than 25. 
A second station a has been operated n years. Let 

n < N and 

9 The letter 5 (Secular Station) has, of course, another meaning from that in the 
previous paragraph dealing with Helmert's and Abbe's criteria. 



COMPARISON OF OBSERVATIONAL SERIES 141 

the n years lie within the period of N years. The station S is 
called a "normal" or " secular" station. Let the n-year average 
temperature at the station a be 

i(a, n) 
and the same n-year average at the station S be 

i(S, n) 
Then the difference d is 

d = i(a, n} - i(S, n) 

If this difference is assumed as quasi-constant, the short period- 
average <(a, n) can be reduced to the normal period N for the whole 
of which only the station S has been operated. The reducing 
equation reads: 

l(a, N} = i(S, N) +d 



i(a y N) is the average temperature at the place a reduced to the 
normal period. 

The temperature series of the following table may serve as an 
example: It is assumed that the stations New York (F), and New 
Haven (H) have been operated simultaneously only in the follow- 
ing January months: 

y, F H, F 

January 1912 ...................... 23.2 21.2 

1914 ...................... 31.1 29.2 

1917 ...................... 32.1 30.6 

1918 ...................... 21.3 19.8 

1920 ...................... 23.8 21.4 

Average of 5 years ................. 26.3 24.5 

d = l(H 9 5) - i(Y,S) = - 1.8 F 

If New York is the normal station which has been operated 
during the entire 10-year period 1911 to 1920, we get the equation: 



) = l(Y 9 N)+d 
and here 

l(H, 1911 to 1920) = i(Y, 1911 to 1920) - 1.8 F 
According to Table 35 : 

i(Y, 1911 to 1920) = 31.0F 



142 METHODS IN CLIMATOLOGY 

Therefore the reduced value 

i(H, 1911 to 1920) = 29.2F 

In reality, the station at New Haven has also been operated 
over the entire 10-year period. The observed average is by chance 
identical with the calculated value. 

It may be remarked that the 5 -year average at New Haven 
is nearly 5 F lower than the reduced one. Therefore the 5 -year 
average cannot be compared with a 10-year average of another 
station. 

The reduction to equal periods is fundamental for all climato- 
logical and climatographical purposes. 

XII. 3. b. The Reduction of Precipitation Series: 
Method of Quotients 

For series representing rainfall, the procedure of reduction to a 
normal period is analogous, but ratios are used instead of differences. 

A normal (secular) station 5 has been operated over the normal 
period of N years. Another station a, located at a small distance 
from S, has been operated n years, so during these n years simul- 
taneous observations have been conducted at the two stations. 

The nomenclature is (partly repeated) : 

N = normal period of the Secular Station S 
n = short period during which the station a has been op- 

erated (n < N}. 
P(S, N) = Normal amount of rainfall at 5 during the nor- 

mal period N 

p(a, n) = average rainfall at a during the short period n 
P(S, n) = the same for the normal station 
p(a, N) = average annual rainfall at a, reduced to the 
normal period 

p(a, N} is the unknown value. Then under the assumption of 
quasi-constancy of the ratio q: 



P(S, n) P(S, N} 

Therefore 

p(a, N} = q X P(S, N) 

To take an example from Table 36A : Let Boston Weather Bureau 
(B) be the "normal" station, operated during the entire period 



COMPARISON OF OBSERVATIONAL SERIES 143 

1911-1920. At the Cambridge Waterworks, Waltham, Mass. (W), 
precipitation might have been measured for only five years, with 
the following results at the two stations: 



July 
1913 


B 
2.69 


W 
3.29 


1914 


2.64 


2.46 


1917 


1.10 


0.93 


1918 


2.64 


2.99 


1920 


1.56 


1.78 









sum 10.63 11.45 

P(W, 5 years) = 11.45 inches 

P(B, 5 years) = 10.63 inches 
and 

p(W, 5 years) = 11.45 = p(W, 10 years) 

P(B, 5 years) 10.63 P(B 9 10 years) 

According to Table 36, the 10-year average for Boston, Mass., 
is 3.96 inches; therefore: p(W,lQ years) = 1.08 X 3.96 inches 
= 4.28 inches. As is seen from Table 36, the true 10-year average 
for Waltham is 4.10, so that the difference 

observed calculated = 0.18 inches 

The error is 4.39%. It is a rather good agreement if one con- 
siders that the actual 5 -year average for Waltham was 2.29 inches, 
against a true mean of 4.10 inches. 

The reduction, reasonably applied, always yields reasonable 
results, while the comparison of different periods leads to false 
inferences. 

XII. 3. c. Physical Explanation of the Method of Quotients 

A short discussion must be added to explain why "ratios" 
instead of differences are used for the reduction of rainfall amounts. 

There are two stations, a and 6, with the average rainfall 
amounts p(a) and p(b). The assumption is made that p(b) > p(a). 
This may be expressed as follows: p(b} = p(a) + A(a). 

In a given year, a precipitation greater or smaller than normal 
may occur at a, that is, k X p(a), where k is a coefficient greater 
or smaller than 1. 

The question is, what amount is recorded at b. 

If the differences were invariable, p'(V) = k-p(a) + &p(a). 



144 METHODS IN CLIMATOLOGY 

Then A(a) would be the constant difference between the 
amounts at the two places. 

This assumption does not appear likely. The average incre- 
ment A/>(a) is caused at the place &, for example, by the contour of 
the region or some other physical feature. It is, therefore, much 
more probable that p(a) and the increment &p(a) are increased 
according to the ratio k. We have therefore to expect at the place 
b the amount kp(a) + kbp(a). 

Numerous examples which prove this consideration could be 
presented; hence it can be taken as certain that the ratios of the 
rainfall amounts at places not too far apart are quasi-constant, 
in contradistinction to their differences. 

XII. 3. d. The Reduction of Other Elements to a Given Period 

From the foregoing, it is clear that averages of temperature 
and precipitation at different places are comparable with one an- 
other only if they are related to an identical period. This postu- 
late is obviously valid for each climatic element, be it primitive, 
derived, or combined. 

There remains the question whether differences or quotients 
have to be chosen. Only a few oft the most important elements are 
analyzed in this respect, namely: 

a) Number of days with precipitation 

b) Cloudiness 10 

c) Duration of snow-cover u 

d) Number of days with fog 

As the differences in the case of these various climatological 
elements yield better results than ratios, or results equally good, 
the method of differences is preferred because it involves easier 
calculating. 

XII. 3. e. Limits of the Method of Reducing Climatological 
Series to a Normal Period 

The basic principle of the " reduction " to a normal period is 
the quasi-constancy of differences or quotients. The mathematical 

" V. Conrad, "Zum Studium der BewSlkung," Met. Zeit., 1927, pp. 87-91. 
11 V. Conrad, M. Winkler, "Beitrag zur Kenntnis der Schneedeckenverhaltnisse in 
den dsterreichischen Alpentendern," Gerland's Beitrdge zur Gcophysik, XXXIV, 473-511. 



COMPARISON OF OBSERVATIONAL SERIES 145 

meaning of the word " quasi-constancy " is that the variability of 
differences and quotients is much smaller than the variability of 
the elements themselves. From the example in Table 35 may be 
derived : 

v (diff .) = 0.43 
v (temp. F) = db 4.89 
v (temp. H) = 5.08 

As was said earlier, v (diff.) is about a twelfth of the variability 
of temperature. 

Obviously, the average variability of the differences (quotients) 
increases with increasing distance between the two places. The 
same effect occurs if one of the two places is shifted beyond a 
climatic divide. In both cases, the variability of the differences 
becomes greater and greater; continuously in the first case, and 
eventually discontinuously in the second. 

Apparently, the quasi-constancy disappears when the vari- 
ability of the differences (quotients) equals the variability of the 
element itself; then a reduction to a normal period is no longer 
reasonable. That is the theoretical limit of the reduction, and is 
expressed by the equation : 

v (diff., quot.) = v (element) 

If v (diff. or quot.) is called v(R) and v (element) is called v(E) 
then generally this equation is valid : 

v(R) = k-v(E) 

Only if k < 1 is a reduction theoretically admissible. 

The question is, at what value of k should the practical limit of 
reduction be assumed? Such limits must be determined em- 
pirically. In the writer's experience, the reduction should be 
made only when 

k ^ 



This result is, of course, valid for all elements. 
The increase of v(R) has been represented by Hann by means 
of an empirical, analytic equation, which reads: 

V (R) = a + 6-AE + c-bh 

where a, b, c, are constants, AE the distance between two places, 
and A/& the difference of height. 



146 METHODS IN CLIMATOLOGY 

This equation is mentioned here because it appears in many 
manuals, and also because it can be misleading. 
At the limit: 

AE - Aft = 

the differences are related to the element of the " normal station" 
itself. Then, however, v(R) = 0, 

This does not occur in the equation above, because the con- 
stant a ^ O. 12 

In Chapter IV, 2, phenomena of decay are discussed and an 
example is given to show how the variation of v(R) for cloudiness 
can be represented by an analytical equation which exactly ful- 
fills the boundary conditions. 

Lack of space prevents a fuller discussion of the problem; but 
the above hints should suffice for further investigations. 

Reduction to a normal period has to be combined with an 
examination of the relative homogeneity, the equally important 
investigation into the nature of a climatological series. Both 
procedures require the computation of deviations. 

Finally, it may be remarked that, in the opinion of many 
authors, very long series are comparable with one another without 
being reduced to the same periocL Practically, one may be lucky 
enough to deal with series which happen to be comparable. It is 
perhaps more likely with non-reduced long series than with short 
ones. 

In any case, however, especially with long series, the examina- 
tion of relative homogeneity is unavoidable. 

XII. 3. f. Additional Note. The Length of a Normal Period 

It is frequently asked, how long a normal period should be. 
This depends upon three main conditions : 

1) How many years of incontestable observations of the 
normal station or stations are available. 

2) How long the whole period must be in order to include 
the periods of the series which have to be reduced. 

3) How great the variability is of the element in question. 

Because of these conditions it is not possible to define a priori 
how many years make a normal period. 

12 For more details see V. Conrad, "Zum Studium der Bewdlkung," Met. Zeit., 
1927, p. 87. 



COMPARISON OF OBSERVATIONAL SERIES 



147 



Table 40 gives examples of the enormous variability of rainfall 
at Maiden Island (4S). In view of this fact, perhaps 50 years 
may not suffice to report, even approximately, the variations 
possible at this place. On the other hand, the variability of 
temperature on this tiny island in the equatorial belt of the Pacific 
Ocean, is nearly negligible. With regard to temperature, a normal 
period of 2 years would be sufficient. 

Some scientists are of the opinion that the normal period should 
include at least one "Bruckner-cycle," i.e., about 35 years; that is, 
a normal period should not be shorter than 35 years. The follow- 
ing considerations lead perhaps to the same result. 



1*0 
140 
120 
LOO 



10 20 30 40 50 60 70 80 90 100 

YEARS 

FIG. 27. Increase of range of the extremes of temperature with length of period 

Birkeland and Frogner 13 have shown how the range of the 
extremes of temperature and rainfall increases with an increasing 
period of observation. Their investigation indicates that this 
correlation (extreme range-length of period) can be computed 
from the Gaussian law of errors. An example is given for the 
range of the extremes of temperature in Fig. 27. The range of a 
period of 10 years is taken as the unit; this unit range is increased 
by 20% during a further 10-year period of observation. 

The increase from 40 to 100 years is but about 20% as a conse- 
quence of the asymptotical trend of the curve. The increment 
from 100 to ISO years is 3%, and 2% for the following 50 years. 
These theoretical considerations are valuable hints, when making 
a reasonable choice of the length of a normal period, if a long series 

11 B. J. Birkeland, E. Frogner, "Die extreme Variabilit&t der Lufttemperatur," 
Met. Zeit., 1935, p. 349. 



148 METHODS IN CLIMATOLOGY 

of observations is available at all. But it is more important to 
take that portion of the long series which is incontestably homoge- 
neous than to strive for a " normal period" as long as possible. 

The upper limit of the length of a normal period should be 
dictated rather by the condition that the periods of the other series 
be included than by the intention of extending the normal period 
uncritically to infinity. It is undesirable for obvious reasons that 
a normal period of 100 years be taken, if the longest of the other 
series do not cover more than perhaps 20 years. 

These are some of the ideas which should be considered when 
choosing a " normal period. " In any case, its length should ex- 
ceed 10 years, because otherwise the conclusions from statistical 
methods are no longer useful. From all points of view, the 
length of a normal period should be about 25 to 35 years. 14 

XII. 3. g. Interpolation of Missing Observations 

The quasi-constancy of differences or quotients, offers the 
possibility of interpolating missing or false values. Curve parallels 
for the variations of the respective element at two and more sta- 
tions are often useful, but numerical differences, and quotients in 
particular, should be preferred* This is the more essential as 
these calculations must be made in order to investigate homo- 
geneity, etc. Systematic and random errors are revealed by these 
simple methods, and can often be corrected to a certain degree. 
Important corrections should be made if possible by means of 
at least two pairs of stations, each of which includes the dubious 
station. 15 

Only a full comparison ot the synchronous climatological series 
used yields reliable results. 

XII. 3. h. Coherent and Incoherent Climatic Regions 
and Their Statistical Evidence 

The increase of the variability of differences (quotients) with 
increasing distance between the places compared was mentioned 
in connection with the method of reducing climatological series to 
an identical period. Although previously not emphasized, the 

14 The International Meteorological Organization adopted the period 1901-1930 as 
a '* standard period" for climatological normals (Warsaw 1935). See the Meteorological 
Glossary (London, 1939), under "Normals." 

16 For details, see V. Conrad, "Die klimatoiogischen Elemente," p. 119, chapter 7, 
The interpolation of missing observations. 



COMPARISON OF OBSERVATIONAL SERIES 149 

increase of the variability of differences occurs more markedly if 
the difference in elevation between the places selected is increased. 
The increment of variability is a continuous process under other- 
wise equal conditions. This has been shown in a study of cloudi- 
ness. 16 As was said before, one may represent the increase of 
v(R), the variability of the differences (quotients) with an exponen- 
tial equation. The values calculated with this equation are in 
close agreement with observation. Hann's formula, even if not 
in accordance with actual conditions in the immediate surround- 
ings of the normal station, shows too that the differences increase 
continuously. Since this behavior holds for two climatological 
elements, one can assume that the same is true for other closely 
related elements. 

This problem can be inverted in so far as the kind of variation 
of v(R) can be taken as a climatological characteristic of a region. 

A region within which the variability of differences (quotients) 
of elements at two places increases continuously with increasing 
distance (difference in elevation) is called climatically coherent. 
Lines or strips at which the variability of differences (quotients) 
varies discontinuously are called climatic divides. Therefore re- 
gions on different sides of a climatic divide are called climatically 
incoherent. 

These concepts, derived from the variability of the differences 
(quotients) of elements at two places, are of major importance for 
recognizing the climatical structure of a region, and are funda- 
mental for conclusions regarding the geographical distribution of 
the climatic elements. Consequently, the short and long range 
forecaster is most concerned with these ideas. A forecast fully 
valid for climatically coherent regions fails if it is extended to an 
adjacent, but climatically incoherent, region. 17 

These considerations, coupled with those of homogeneity, lead 
to the method of correlation, most important in so many climato- 
logical problems, which will be discussed in the following inter- 
mediate chapter. The behavior of the correlation factor may be 
decisive whether or not climatic regions are coherent. This idea 
offers a great field for future investigations. 

16 V. Conrad, "Zum Studium der Bewolkung," Met. Zeit., 1927, pp. 87 ff. 

17 See also I. I. Schell, "On the Use of Climatically Coherent Areas in Seasonal 
Foreshadowing, " BAMS, vol. 23 (1942), pp. 182-183. 



INTERMEDIATE CHAPTER (C) 
CORRELATION 

C. 1. LINEAR CORRELATION 

MATHEMATICALLY, the ' ' quasi-constancy " of, for example, 
differences between average temperatures at two places 
means that a certain relationship exists between the tem- 
perature series at the two places. The ratio between the variabil- 
ity of the differences and the variability of the element itself is the 
measure of the association of the two temperature series. 

This measure is specifically climatological and should be stated 
in terms of those measures offered by the statistical method of 
correlation. A generally valid measure of the degree of association 
of two series in the simplest case can be easily established. 
Two series may consist of the following elements : 



X n 
and * 

Fi, F 2 ; - F n 

The elements of these series have to be compared with one another 
in pairs, i.e., X\ with Y\\ X% with F 2 ; etc. 

Because of the inequality of the arithmetical means (Jt and F), 
the direct comparison is possible only by introducing deviations : 

Xi - = Xi and F< - ? = y< 

There is no doubt that the sum of the products Xiy* is a measure 
of the degree of association of the two series. 

If the number of the elements is great and there is no relation- 
ship between the two series, the products xty* are random numbers 
which may vary between certain definite limits. Then it must 
be possible to find for each product another of equal magnitude 
and opposite sign. (See the analogous conclusion with Abbe's 
criterion.) 

This means that the sum of all products is zero if no association 

f=n 

exists. Furthermore the value of #,-y< depends, obviously, on 

i=l 

the number n of the elements of the series. Therefore the sum 
of the products has to be divided by n, in order to free it of the 

150 



CORRELATIONS 151 

influence of the arbitrary number of elements and we get 



There are two further difficulties in comparing the Xi series and the 
yi series: 

1) The different elements may be expressed by different 
units. 

2) The scattering of the deviations of the two series is 
different. 

If the foregoing expression is divided by the products of the 
standard deviations of the two series, it is reduced to equal scatter- 
ing and the different units are eliminated at the same time. 
According to what has been said previously, 

- 

and * 





Therefore the measure of the degree of association of the two 
series is: 



n 



fejc, 2 _ 2y7 2 
\ n ' n 

or finally 



which appears often in the form 



The expression r xy is called correlation coefficient. Obviously 

r xv = + 1 

if the functions /(#;) and/(y t ) are identical or differ by a constant 
factor. In other words, the two series run parallel to each other. 



152 METHODS IN CLIMATOLOGY 

If the trends of /(#;) and/(y;) are in exact opposition: 

as it is clear from the equation for r xy . 

These are the boundary values that occur if the association of 
the series is perfect and can be expressed by means of a normal 
analytical equation. As stated before, r xy = if no association 
exists between the two series. 

On the other hand, it is clear that a correlation coefficient 
greater or less than zero can often result by chance, even if no 
association between the two series exists, when only a small 
number of observations is available. In this case, it is unlikely 
that 2>Xiyi will become zero ; consequently the correlation factor r 
differs from zero also for unrelated series. Even with a greater 
number of elements r will be falsified in a similar way. 

It can be said, however, that then the true correlation coeffi- 
cient r, which would result from an infinite number of elements of 
the two series, lies with equal probability within or without the 
limits r +/ and r /. The value/ is the probable error mentioned 
previously (III. 6). It is: 

f = 0.6745 1 



A/n 

With an increasing number of elements the probable error 
decreases more and more, approaching zero. (See Table for 
1/Vn in Appendix II.) 

The smaller / is, the more significant is the respective correla- 
tion coefficient. K. Pearson gives the rule of thumb that correla- 
tion coefficients should be assumed as real only if they are at least 
6 times greater than the probable error. According to the author's 
experience, this ratio is very useful in avoiding false conclusions. 

On the other hand, the limit r xy ^ 6f is too high, according to 
other authors, for estimating the reality of correlation-factors. 

In this connection, the opinion in the Meteorological Glossary 
(London, 1939, p. 54) is quoted: 

Unless it is confirmed by physical reasoning or other independent 
evidence, a correlation coefficient should not be accepted as significant, 
unless it exceeds three times its probable error, in which case the odds in 
favor of significance are 20 to 1. If a number of trial correlations are 
made, the chance of obtaining a single large coefficient is obviously 
greatly increased, and such an isolated coefficient should not be ac- 
cepted, unless it is four or five times its probable error. 



CORRELATIONS 153 

In the main, it is of greatest importance to calculate not only 
the correlation factor but also the probable error. Only then is it 
possible to estimate the significance of a correlation factor. 

Generally, correlation coefficients considerably smaller than 
0.5 have to be estimated most critically. 1 

On the other hand, cartographical representation of correlation 
factors are very instructive. If a great region is covered by one 
sign and another region by the opposite sign and the two regions 
are systematically separated by a transition zone of mixed signs, 
in this case small correlation coefficients can also lead to further 
successful investigations. 

If the correlation of a pair of series covering a great number of 
years shows a certain sign, the series should be divided into parts. 
For each of them, a correlation coefficient should be computed. 
If the signs of the single coefficients are identical, a certain sig- 
nificance even of small coefficients cannot be rejected without 
argument. 

C. 2. EXAMPLE OF CALCULATING A CORRELATION 



The average air pressures in millimeters of mercury (Xi) on the 
one hand, and the numbers of hours with bright sunshine (F) on 
the other, are known for 10 months of February at a place in the 
temperate latitudes. % and T are the arithmetical means, x> and 
yi the deviations. (Table 41.) 

The opinion is frequently expressed that high pressure corre- 
sponds to fair, sunny weather in temperate latitudes. This as- 
sumption is examined by means of the values X iy Yi in columns 2, 
3 of Table 41. The correlation coefficient is below 1/2, and is 
not even double its probable error. 

Therefore we have to conclude that a series of 10 pairs of 
average values does not offer any reason for assuming an associa- 
tion between air pressure and synchronous hours of bright sun- 
shine, as far as averages (sums) and the region in question are 
concerned. 

The plus sign ( +) could indicate the tendency in the sense of the 
opinion mentioned; it would be significant only if it reappeared 
also with other equivalent pairs of series. 

1 Further remarks on this problem will be found at the end of C. 4. For an excellent 
model for this procedure, see Ellsworth Huntington, Tree Growth and Climatic Interpreta- 
tions in Quarternary Climates (Washington: Carnegie Institution, 1925), p. 161 ff. 
Attention is called also to the interesting "Dot Charts" of the relation between annual 
growth of sequoias and rainfall. 



154 METHODS IN CLIMATOLOGY 

TABLE 41. EXAMPLE OF CALCULATING A CORRELATION 

(1) (2) (3) (4) (5) (6) (7) (8) 

No. Xi Yi xi y\ Xiyi x* y<* 



1 


43.7 


60 


-3.0 


-26 


+ 78.0 


9.0 


676 


2 


51.5 


133 


+4.8 


+47 


+225.6 


23.0 


2209 


3 


50.6 


72 


+ 3.9 


-14 


- 54.6 


15.2 


196 


4 


47.9 


82 


+ 1.2 


4 


- 4.8 


1.4 


16 


5 


44.6 


119 


-2.1 


+33 


69.3 


4.4 


1089 


6. . . 


.. .47.2 


52 


+0.5 


-34 


- 17.0 


0.3 


1156 


7 


52.9 


81 


+6.2 


5 


- 31.0 


38.4 


25 


8 


41.0 


54 


-5.7 


-32 


+ 182.4 


32.5 


1024 


9 


37.1 


83 


-9.6 




+ 28.8 


' 92.2 


9 


10 


50.1 


129 


+ 3.4 


+43 


+ 146.2 


11.6 


1849 



















46.7 Y = 86.5 

*0* +484 



r _ 



V228 X 8249 



0.6745 = 0.187 



H 




Note: For calculating/ see Appendix II. * 

To avoid a misunderstanding, it should be expressly remarked 
that the correlation factor does not indicate anything about the 
causal association of two series of numbers representing the varia- 
tions of two geophysical phenomena. The indications of the 
correlation coefficients are only of formal nature. This fact cannot 
be sufficiently emphasized. Two series can show a high correlation 
coefficient, because of a systematic observational mistake, or 
because the association is physically well justified, as, for instance, 
for synchronous values of air pressure at two places a few miles 
apart. Therefore it is not sufficient merely to calculate the corre- 
lation factors. If they are known, the really scientific investiga- 
tion begins with the physical explanation of the correlations 
computed. 

In the example of Table 41, the explanation is obvious. The 
tendency toward the occurrence of precipitation is evidently 
smaller with high pressure than with low pressure. But above a 
plain, especially above a city, fog is much more frequent with high 
pressure conditions in winter. The final result is that a direct 
connection between air pressure and the number of hours with 
bright sunshine is lacking in this special case. 



CORRELATIONS 



155 



C. 3. SIMPLIFICATIONS IN COMPUTING CORRELATIONS 

The number of applications of the method of correlation is un- 
limited in climatological problems; so some simplifications of the 
technique of calculation may appropriately be indicated here. 

The computation of the deviations (xy) is sometimes weari- 
some. If the simple definitions previously mentioned are con- 
sidered, the following identities result: 

Xi = Xi - X and y t = Y* - Y 



E 

1 



; = nX 



Therefore 



and 



and 



and 



T = n? 



i = n? 



E 3cy.- = E ^i^ - nX? 



(see footnote 2 ) 



L v 2 _ V P 
Xi 2^ V/ 


^ Y^2 V Y 2 *t V2 
\ i A^ Z^-^-i ^-^ L 


i i 


1 


n n 


n 


E y* 2 = E ( 


Yi F) 2 = E FV 2 wP 2 


i i 


i 



- F) (Definition of x< and 



Then 
Because 
and 
and 
we get: 



i i 

The further transformations are analogous. 



nJPF 



156 



METHODS IN CLIMATOLOGY 



These expressions substituted in the original equation : 



n 

x<y< 

1 



yield 



X* - wl 2 



) 



F< 2 - 



This formula uses only the quantities of the original series, avoid- 
ing the calculation of deviations. 

If we look at the numbers in columns 2 and 3 of Table 41, the 
original items, a further trouble is apparent; the large numbers 
with which one has to calculate. This inconvenience also is 
easily avoidable, if similar, simple considerations are made as above. 
Each number of the two series is reduced by a number which 
equals the minimum of the respective series. That is 37.1 in the 
series of column 2, and 52 in the seYies of column 3. Thus we get 
small numbers for the calculation quickly and easily. 

TABLE 42. CALCULATING A CORRELATION BY MEANS OF THE 
ORIGINAL NUMBERS REDUCED BY A CONSTANT VALUE 



No. 



X/F/ 



YS* 



1 


6.6 


8 


52.8 


43.6 


64 


2 


14.4 


81 


1166.4 


207.4 


6561 


3 


13.5 


20 


270.0 


182.2 


400 


4 


10.8 


30 


324.0 


116.6 


900 


5 


7.5 


67 


502.5 


56.3 


4489 


6 


10.1 


00 


0.0 


102.0 


o 


7 


15.8 


29 


458.2 


249.6 


841 


8 


3.9 


2 


7.8 


15.2 


4 


9 


0.0 


31 


0.0 


0.0 


961 


10 


13.0 


77 


1001.0 


169.0 


5929 















JP' 9.6 ?' = 34.5 VXt'Yi' =3782.7 VXi'* 1141.9 ZFi 7 ' = 20149 
n - 10; n%'?' - 3312.0; nX<'* - 921.6; wlY' 11902.5 



470.7 



' - nf'*) 1347.9 



+ 0.349 



CORRELATIONS 157 

The example of Table 42 is identical with that of Table 41. 
The correlation between pressure (Xi) and sunshine hours (F) is 
calculated with these original numbers instead of the deviations 
Xi and y^ Furthermore, Xi and F are reduced by the minimum 
values of the respective series. 

The new nomenclature is : 

Xi - 37.1 = X^ and F, - 52 = F/ 

The other details of Table 42 are understandable without ex- 
planation. The results of the calculations in Tables 41 and 42 are, 
practically, identical. 

C. 4. REGRESSION EQUATION 

The deviations from the arithmetical mean Xi and yi (see C. 1.) 
can be plotted in pairs on a graph paper representing a Cartesian 
system of coordinates with the origin x = and y = 0. The 
points Xiyi can be scattered in such a way that they cover the 
quadrants of the coordinates with the greater uniformity the greater 
the number n of the pairs of deviations is. This is not of interest 
here. On the other hand the dot-diagram can show a crowding of 
the points, indicating a linear proportionality in the simplest. 
(See Fig. 28.) Then we can try to determine a straight line y = bx 
which passes the origin and is fitted to the point-cloud as well as 
possible. 

If x should be an analytical linear function of y then y bx 
would be exactly zero. In this problem, however, this is not valid, 
and the quantity b must be determined so that it makes the differ- 
ence y bx as small as possible. 3 Since the signs of the deviations 

1 The reader may profit by another very expedient method of explaining, kindly 
communicated to the author by L. W. Pollak, Dublin, Ireland, in a letter. I quote: 

"We have observed n pairs of phenomena Xi, Yi. Forming the means of Xi and 
Yi respectively, we obtain 

(i) 



n n 

The values X t Y define in the graphical representation of our Xi, Yi a point M which as 
the 'centre* of the dot diagram plays an important role. 

14 Further, we compute the deviations of the Xi and Yi from their arithmetical means, 
obtaining 

Xi -Xi- Xi - Xi -X, yi - Yi -?i-Yi-Y (2) 

Plotting the Xi, yi means nothing but shifting the origin of the codrdinate system of the 
Xi, Yi dot diagram to M. 

"Now we intend to approximate our deviations #,, yi (in the codrdinate system with 



158 METHODS IN CLIMATOLOGY 

do not play any role in this problem, only their absolute amounts, 
b should be chosen so that the sum of the squares of y bx is as 
small as possible. This is reached if 2(y bx) 2 becomes a mini- 
mum. The condition is fulfilled, if 



and 

22 (- x)(y - bx) = or Sry = 
Therefore 

. 



n 

The equation y = bx contains the arbitrary assumption that x is 
the independent variable. 
One can also write : 

x = b'y 
and 

6'=i 



M as origin) by a straight line 

y = bx + d (3) 

so that the sum of the squares of the distances of each dot from our straight line, meas- 
ured in the direction of the /s, is a minimum. 
"Therefore 

S[y< - (bxi + d)J = Min. (4) 

and accordingly 



dd 
or 



The second equation of (6) gives directly: 

nd 



and 

rf = 

as S#i = and 2yi = 0. (The sum of the deviations from the arithmetical mean, re- 
garding the signs, is zero). Our straight line required passes through the origin or the 
'centre 1 Af." 



CORRELATIONS 159 

If we consider that the standard errors 




n 
and the correlation coefficient 





TABLE 43. ANNUAL VARIATIONS (DEVIATIONS) OF THE LEVEL OF VICTORIA NYANZA 
(0N, 32E), (xi inches) and of the Annual Sunspot Numbers (y) 

(After Sir Napier Shaw) 



(1) 


(2) 


(3) 


(4) 


(5) 


(6) 


Year 


x\ 


yi 


Xt* 


yS 


xtyi 


1902 


-18 


-35 


324 


1225 


630 


1903 


5 


-16 


25 


256 


-80 


1904 


10 


2 


100 


4 


20 


1905 


7 


23 


49 


529 


161 


1906 


21 


14 


441 


196 


294 


1907 


13 


22 


169 


484 


286 


1908 


2 


9 


4 


81 


18 


1909 





4 





16 





1910 


- 7 


-21 


49 


441 


147 


1911 


-15 


-34 


225 


1156 


510 


1912 


-19 


-36 


361 


1296 


684 


1913 


-11 


-39 


121 


1521 


429 


1914 


-10 


-30 


100 


900 


300 


1915 


- 4 


7 


16 


49 


-28 


1916 


7 


17 


49 


289 


119 


1917 


27 


64 


729 


4096 


1728 


1918 


19 


41 


361 


1681 


779 


1919 





24 





576 





1920 


- 5 


- 2 


25 


4 


10 


1921 


-13 


-15 


169 


225 


195 


2 






3317 


15025 


6202 















mean 166 751 310 

r xv = - ^ = + 0.88 <r, = Vl66 = 12.9 

b xy = r- = 0.414 x = 0.414? <r y = VlSl 27.4 

cr v 

6 tf x = r = 1.869 y = 1.869* 



160 METHODS IN CLIMATOLOGY 

then 

T 0*V J 7/ ^JC 

= r and Z>' = r 
<r x or y 

Therefore we get : 

(Ty - 0"x 

y r x and # = r y 

G x Gy 

These equations have been called regression equations by Sir 
Francis Galton (1886); 6(6') is called the regression coefficient. 

An example (Table 43) 4 serves as an illustration. 

Let %i denote the deviations (inches) from the average level of 
Lake Victoria (Victoria Nyanza) in Africa, and let y denote the 
deviations from the mean of the annual sunspot numbers (period 
1902-1920). The values of columns 2 and 3 in Table 43, # and y,- f 
are plotted on graph paper. The values # mean the variations of 
the lake level (horizontal axis) and the y the variations of the sun- 
spot numbers (vertical axis). This dot diagram (Fig. 28) shows 
clearly that there is a rather strong correlation between the two 
phenomena. 

The two regression equations mentioned above are represented 
by two different straight lines irf Fig. 28. The smaller the angle 
between the two regression lines, the higher the degree of associa- 
tion and the greater the value of the correlation-coefficient r. If 
r = 1, the two regression lines coincide. If r = the one line 
is identical with the X-axis and the other with the F-axis (y = ; 
x = 0). 

As seen from Table 43, the correlation coefficient is high, and 
from Figure 28 it is clear that the angle between the regression 
lines is small in consequence of the strong association. Such cases 
are very rare in the realm of meteorology and climatology. The 
method of the curve-parallels is often misleading, and dot-charts 
are by far preferable to this method. Calculating correlation- 
coefficients and probable errors is never wasted time if the dot- 
chart is somewhat encouraging. 

The correlation coefficient, however, and the probable error, 
do not yet entirely satisfy the climatologist (as we have mentioned 
before). The critical estimation of the true value of a correlation 
coefficient is difficult. W. H. Dines says: 5 

4 Taken from C. E. P. Brooks, Climate through the Ages (New York, 1926), p. 415 ff, 
and Sir Napier Shaw, Manual of Meteorology, I, 284, Fig. 116 and table on the same 
page. The continuation of the example concerning the partial correlation is also taken 
from C. E. P. Brooks' discussion of the problem. 

6 The Computer's Handbook (Met. Office, London, 1915), p. V 39. 



CORRELATIONS 



161 



No precise classification of correlation coefficients can be made. As- 
suming that they depend on as many as 50 independent cases, one may 
say roughly that values under 0,30 are hardly significant, values between 
0.30 and .70, prove a moderate connection; values between .70 and .90, 
a close connection, and values over .90, a very intimate connection indeed. 

It is also apparent that, if the coefficient is very high, fewer observa- 
tions are necessary to establish it, since 1 r 2 is then small. 6 




-20 




_i , LAKE 

\B 3) LEVEL 



ZQ 



40 



60 



FIG. 28. Dot-chart and regression lines of the deviations from mean annual 
level of Victoria Nyanza [0 (Eq.), 32E] and from the mean number of sun- 
spots. The mean level of the lake and the mean number of spots are taken 
as axes. (After Sir Napier Shaw) 

From this classification, the coefficient of the example in Table 
43 is only " close, " and C. E. P. Brooks expresses this fact by 
saying that " there is a good, but by no means perfect, agreement 
between the variations of the level of Victoria Nyanza and sun- 
spot numbers/' We can assume that " variations of level are 
'caused' partly by variations of sunspots and partly by other 
factors, which for the sake of illustration we shall suppose to be 
independent of sunspots." This portion of the variations of level 
is, however, 1 - r 2 = 1 - (0.88) 2 = 0.23, or 23%. 

C. 5. PARTIAL CORRELATION 

In the foregoing, the problem of two variants which show a 
certain degree of association has been discussed. This case, 
6 See probable error. 



162 METHODS IN CLIMATOLOGY 

apparently, is the most important for the climatologist. The 
general problem, however, is that there are m variants, mutually 
dependent upon one another. The question is, how and with 
what accuracy can x be represented by the other variants, y, z, 

The method starts from the aforementioned correlation coeffi- 
cients and analyzes the complex influence of each of the series of 
variables upon x, so that the partial influence of the single variables 
upon x is separated one from the other. Therefore we speak of 
" partial correlation/' 

If, for instance, three variants are associated one with another, 
the problem is to find how the second variable is correlated with the 
first, when the third is constant, and so forth. 7 

Now we return to the example of Table 43, which indicates 
that the variations of the level of Victoria Nyanza (x) are closely 
associated with the synchronous variations of the numbers of 
sunspots (y). 

In this example, especially, the need of a connecting link in the 
form of a third variant is evident: this is the rainfall. Its varia- 
tions (z) are known, and the correlation between z and x is indeed 
high : 

r x . = -.92 

The coefficient between rainfall variations (z) and the variations 
of sunspot numbers (y) is: 

r yz = + 0.80 
Theoretically two alternatives exist : 

1) The sunspots influence the rainfall, and its variations 
cause the variations of the level of the lake. 

2) There is a direct connection between sunspots and lake 
level. 

One should discriminate between these two alternatives. For 
this purpose, the partial correlation coefficients are calculated. 
We repeat the nomenclature : if 

T X y = the coefficient between lake level (x) and sunspots (y) 
r xg = the coefficient between lake level (x) and rainfall (z) 
r yz = the coefficient between rainfall (z) and sunspots (y) 

7 C. U. Yule, An Introduction to the Theory of Statistics (2nd ed., London, 1912) 
and F. M. Exner, Ueber die Korrelations Methode (Jena, 1913). 



CORRELATIONS 163 

the partial correlation coefficient 8 

is given by the equation 

' XV """" 1X2 ' ' UZ 

r 
xy.z 



From the example we get : 

0.88-0.92-0.80 
'*"" = 0.39-0.60 = + ' 61 

The meaning of r xy . z is that " the effect exerted through rainfall " 
is "eliminated." In other words, the correlation between x and y 
is examined at constant rainfall. The relatively small correlation 
coefficient r xy . 2 shows that the major part of the effect on the level 
of the lake is caused by rainfall ; that corresponds well to common 
sense. On the other hand, " there is still an appreciable effect 
due to some cause independent of rainfall, most probably rate of 
evaporation." 

The expression for r xv . z has been given. The two other coeffi- 
cients are easily derived from this equation. We exchange y for z, 
z for y, and get from 



and an exchange of x for y, which in turn yields, 

fyz.x from r xe . y 

8 D. Brunt, The Combination of Observations, (Cambridge, England, 1931), p. 175, 
gives an interpretation of the partial correlation coefficient which may be quoted here 
as a very valuable addition to the above-mentioned explanation. For the convenience of 
the reader, Brunt's symbols are replaced by those used here in the text. I quote: 

"We further require to know the correlation between x and y when each has been 
corrected for the correlation with z. This is written r xl/ .,, indicating that the effect of 
the variable 2 is eliminated. Since the regression equations of x and y on z are 

x r x , 2 = 0, y r y , 2 = 
<r g <r f 

fxy.i is the coefficient of correlation between 

x r xv ~ 2 and y r v , ~ z." 



164 METHODS IN CLIMATOLOGY 

Thus we get three partial correlation coefficients : 

T xy ?xz ' fyz 



?xz.y 

fyz.X == 

The respective standard deviations are : 

(fx.yz == 0W-1 f xy'\- 



The following six regression coefficients correspond to the three 

partial correlation coefficients. 

t 

+ 

h Y (Tx ' vz h - r . (Tx ' vz 

Uxy.z ' xy.z ^xz.y ' xz.y 

Gy.xz &z.xy 

h - r (7v ' xz h - r (Ty ' xz 

u yx.z 'xy.z u yz.x fyz.x 

Gx.yz Gz.xy 



h v Vz.xy , _ Vz. 

Uzx.y 'xz.y Uzy.x 'yz.x* 

&x.yz ffy. 

Finally, the regression equations read : 



z = b xz .-x 



The example mentioned above shows that the method of the 
partial correlation is well suited for discriminating between the 
individual influences of the different variables. Therefore the 
method of partial correlation is much more than a kind of repre- 
sentation: it can be a successful method of investigation. For 



CORRELATIONS 165 

want of space, these remarks about linear correlations must suffice. 
Generally, the common climatological problems can be solved by 
the methods indicated. 

L. W. Pollak's application of ' ( autocorrelation " 9 is of great 
importance for determining hidden periodicities as well as the 
most significant problems of long-range forecasts as far as higher 
statistics are concerned. This method correlates two portions of 
the series with the series itself , which may briefly explain term and 
aim of the method. 

9 L. W. Pollak, "On Cycles of Pressure Especially in the Neighborhood of Symmetry 
Points," QJRMS, vol. LXVI, no. 287 (1940). 



CHAPTER XIII 

GRAPHIC COMPARISON OF CLIMATOLOGICAL ELEMENTS. 
ISOGRAM. MAP OF ISOLINES 

THE PROBLEM is to represent three variables on a two-dimen- 
sional plane. A well-known solution is the representation 
of contour lines on a map. The first variable is the geo- 
graphical longitude, plotted on the abscissa. The second variable, 
the latitude on the ordinate. Points of equal elevation (the third 
variable) are connected with one another by the contour lines. 
This analogy leads to the understanding of the method of repre- 




60 



-if 



FIG. 29. Isopleths representing the annual course of air temperature in the 
high northern latitudes. (After H. Mohn and W. Meinardus) 

sentation. An example of the application to climatological 
purposes is given in Fig. 29. The months are plotted on the X-axis, 
and the geographical latitudes on the F-axis. The monthly tem- 
peratures of the parallels may be known. For 85N, e.g., the 

166 



COMPARISON OF CLIMATOLOGICAL ELEMENTS 167 

numbers read : 

Jan. Feb. Mar. Apr. May Jn. Jl. Aug. Sept. Oct. Nov. Dec. 
C -37 -37 -32J -25 -12 -2 -2 -11 -22 -29 -34 

These temperatures are plotted on the map at the intersections 
of the verticals (indicating the mid-points of the months) and the 
parallels. Now the points of equal temperature are connected by 
curves as in Fig. 29. Along these lines, the temperature is con- 
stant. They are called isotherms for temperature ; (isohyets for pre- 
cipitation, isonephs for cloudiness, isohels for sunshine, etc.) 

The specific representation of three variants of which, e.g., 
the first is the characteristic of an element, the second the time, 
and the third a linear dimension is called isopleth-diagram or 
iso gram. 

Figure 29 takes the place of a table of 7 rows and 12 columns; 
besides, 

1) by interpolation the annual course of temperature can 
be taken from this representation for any latitude be- 
tween 60N and 90N. 

2) The vertical lines mean the middle of the months. Thus, 
by interpolation, the temperature-decrease from 60N to 
the pole can be read off along the vertical for any date 
of the year. 1 

On the other hand, one should bear in mind that the accuracy 
of interpolation is very poor. Be that as it may, the isopleth- 
diagrams are often valuable illustrations of climatological condi- 
tions. The most important and indispensable application of this 
general method is the map of isolines. Isolines are fully analogous 
to contour lines, or to the equipotential lines used in physics. 

From this analogy, laws and rules for drawing isolines can be 
easily deduced. Closed isolines which fully surround a region 
indicate that this is either depressed or elevated. Isolines of 
equal value, running parallel, border an elevated ridge or a de- 
pressed furrow, etc. These principles, translated from the 

1 Attention should be drawn to a special kind of isograms, which show the daily 
course as well as the annual. Irving F. Hand, in "A Summary of Total Solar and Sky 
Radiation Measurements in the United States," MWR, vol. 69 (April 1941), pp. 95-125, 
publishes a great collection of this kind of isograms, which present the best pattern for 
such graphs. They are extremely clear and legible, because they are not overburdened. 
The hours of the day are plotted on the J\T-axis, the months on the F-axis. 

L. W. Pollak gave an original application of the method of isograms, constructing 
iso-stereograms which are certainly of great general value and are also useful for didactic 
purposes (L. W. Pollak, Plastische Tabellen [Plastic Tables], Prague, 1914). 



168 METHODS IN CLIMATOLOGY 

language of the contour-lines into that of climatology, mean for 
instance: high and low pressure region, ridge of high pressure, 
trough of low pressure, saddle between two high pressure regions, 
and similar characteristics for each element which is represented by 
a map of isolines. 

Isolines representing different values cannot cross one another. 
If the trend of an isoline is followed in a given sense, clockwise or 
counter-clockwise, decreasing and increasing intensities (amounts) 
always remain on the same side. Every map of isolines should be 
strictly examined, according to this principle. 2 Isolines should be 
drawn so that the difference of the intensity (amount) is constant 
between any pair of adjacent lines of the same map. Isolines 
which deviate from this general rule should be especially marked 
(e.g. by broken lines, if the regular are full). 

The mutual distance of two isolines is inversely porportional 
to the degree of variation (gradient) of the element. The " den- 
sity " of the isolines is proportional to the intensity of the spatial 
variation of the element. 

The linear interpolation of the intensity (amount) of an ele- 
ment between two points (places), is a necessary expedient in 
default of a better one. But the climatologist should realize how 
rough this approximation is. This ^circumstance on the one hand, 
and the mass of errors with which the average values of the ele- 
ments are infected on the other, should warn everybody against an 
exaggerated accuracy in planning and drawing such maps. It 
must also be emphasized that climatological networks usually have 
wide meshes. Thus, many lines are drawn over a distance of hun- 
dreds of miles, with only two or three points of known values. It is 
regrettable that, sometimes, neither is the homogeneity examined 
nor a reduction to an equal period made. These omissions have 
not deterred students from drawing isotherms for every F, in 
spite of the fact that the actual exactness of these observations is 
about 2 to 3 F. Consequently, the difference of intensity 
(amount) between two consecutive isolines should be weighed 
according to the accuracy of the data at hand, and to the openness 
of the network of stations. 

The map of isolines must never be overburdened. Therefore 
the use of interpolated isolines between those based on observa- 
tions in order to give a better appearance to the map is strictly 
ruled out. Such inserted parallel lines are nothing but an un- 

*See also the short instruction in A. F. Spilhaus and J. E. Miller, Workbook in 
Meteorology (New York, 1942), p. 1. 



COMPARISON OF CLIMATOLOGICAL ELEMENTS 169 

necessary and arbitrary interpolation. Finally, the trend of the 
isolines should be chosen as simply as observations permit. Fanci- 
fully embellished trends mislead the layman, and are little ap- 
preciated by the expert. 

As far as the projection of maps is concerned, Sir Napier Shaw 
and V. Bjerknes arrive at the same conclusions: Shaw recommends 
in his Manual (vol. I, p. 262): ". . . three conformal projections, 
two polar planes extending to 67^, an adjusted conical projection 
for the region between latitude 67^ and 22^, and for the equa- 
torial regions between 22JN and 22|S, projection on a cylinder 
adjusted to be conformal." 

The number of applications of the method of isopleths and 
isograms can be increased arbitrarily, like the number of derived 
elements. Therefore, in the following section and in Part IV, 
only a few hints can be given for making maps of isolines and some 
difficulties mentioned. 



CHAPTER XIV 
METHODS OF ANOMALIES 

XIV. 1. TEMPERATURE DISTRIBUTIONS IN 
MOUNTAINOUS REGIONS 

TEMPERATURE varies with latitude according, to radiation 
conditions and the length of the day; it varies also with 
the distribution of land and water. Moreover, it varies 
with altitude about one thousand times as rapidly as with latitude. 
That is the difficulty. It is fortunate in this respect that about 
three fourths of the surface of the continents does not exceed 1300 
feet, and that 58% is below 660 feet. In relation to these numbers, 
the mountain ranges appear only as disturbances of the continental 
surfaces, as far as the whole surface of the earth is concerned, or at 
least great portions of it. Therefore, the variations of tempera- 
ture with altitude have little influence on the trend of the iso- 
therms, on the whole. t 

To describe the climate of a rrtountainous country is another 
matter; for here the differences in altitude and relief play a decisive 
role, and the climate is determined by these factors. 

XIV. 2. EXAMINATION OF LAPSE RATE 

The variation of temperature per unit of height is called lapse 
rate. It is positive if the temperature decreases with height and is 
negative when it increases. The latter phenomenon is usually 
called temperature inversion. 

Two methods of calculating the lapse rate are used : 

1) The temperature difference of two places at different 
altitudes is computed, and divided by the difference of 
height. The single lapse rates are averaged. 

2) As described in an earlier chapter, temperatures of the 
different stations are arranged according to their respec- 
tive altitudes, then the elevations and the corresponding 
temperatures are averaged within certain intervals of 
altitude. (See IV, 4. b.) Thus a temperature-altitude 
curve is obtained, from which average lapse rates can be 

170 



METHODS OF ANOMALIES 171 

taken for each interval, as well as for the entire difference 
of height. This latter method seems to be better and 
more accurate. The advantage of the first method is 
that a very small number of pairs of stations yield a 
first approximation. 1 

XIV. 3. MAPPING TEMPERATURE CONDITIONS OF A MOUN- 
TAINOUS COUNTRY (REDUCTION TO A GIVEN LEVEL) 

The most common means of representation is by reduction to a 
given level. J. Hann suggested reducing temperatures in a uniform 
manner, independent of season and region, under the assumption 
of a lapse rate of 0.5 C per 100 meters or 3 F per 1100 feet. This 
suggestion should be generally accepted. Although reduced iso- 
therms are not satisfactory in a country with great "relief energy" 2 
and great differences in elevation, at least the comparableness of 
such isotherm maps would result. It is regrettable that uniformity 
is still lacking. Following Hann's rule, the temperature reduced 
to sea level, / , is computed by the formula 

t = (t + 0.005 A)C 

if / is the average temperature in C at an altitude of h meters; or 
to = (t + 0.002743-A)F = (/ + 3/1100-A)F 

if h is expressed in feet. 

Frequently, sea level is not the most advantageous plane, but 
any other level, e.g. the average level of the valley bottom, or 
that of the plain or highland adjacent to the mountains, may often 
yield more plausible temperatures. 

The reduction of temperature when there are great differences 
in altitude is no longer a correction but a method of projection 
onto a plane. This projection should offer: 

1) A survey of the geographical distribution of the tempera- 
ture. 

2) The possibility of interpolating the temperature at a 
place of known altitude from which no observations are 
available. 

1 For numerical comparison between the two methods, see J. Maurer, Das Klima 
der Schweiz (Frauenfeld, 1909), p. 154. 

1 "Relief energy " is the sum of the differences in height along a cross section through 
the region, divided by the length of the cross sections. Obviously, this ratio is zero for 
a plain. Many authors relate the "relief energy" to a certain area and not to a length. 
(H. Wagner, W. Meinardus, Mathemat. Geogr. 1938, p. 349). 



172 METHODS IN CLIMATOLOGY 

At first, this map may be disappointing because it contains 
fictitious temperatures which give a false picture. Only a con- 
version of the temperatures by means of the altitudes (which 
are mostly lacking on the maps) gives the actual temperatures. 

If the picture presented by the reduced isotherms is unsatis- 
factory, what of the possibility of interpolation? Here, too, the 
method does not give good results for winter in temperate latitudes. 
The reason is simple, since temperature inversions are more or less 
common above the valley bottom. This means that the average 
lapse rate is negative, and the average temperatures increase with 
height. It is cold in the valley bottom and much warmer at 
higher levels. The consequences of the reduction of temperature 
to a certain level under these conditions are illustrated by the 
following example: 

The temperature at a place A at the bottom of a valley at 440 m 
reduced to sea level is 4.2C. The actual temperature of a 
place 5, 746 m high, located on the slope, should be interpolated 
from the map. Then 

/ = t - 7.46 X 0.5 C = - 4.2 - 3.7 = - 7.9C 

In reality, observations at *are at hand and the true January 
temperature is 3.7C, which is about 4 C higher than that 
interpolated from the isotherm map. The explanation is simple. 
We have calculated with a uniform lapse rate of 0.5 C/100 m, 
whereas in reality there is an increase of temperature, which can- 
not be considered in the procedure of a uniform interpolation. 

Consequently, one must conclude that the reduced isotherms 
do not yield a suitable representation of the temperature distribu- 
tion in a mountainous region. 

This conclusion, it should be emphasized, applies only in the 
case of a mountainous region. In open land, also in hilly regions, 
where small differences in height are more or less negligible in com- 
parison with the respective horizontal distance, the reduction to a 
fixed level may be accomplished with good results. 

XIV. 4. ACTUAL TEMPERATURES 

Owing to the great disadvantages of the method of reduced 
temperatures, many authors are led to give ''actual temperatures," 
i.e., the true average temperatures as observed at the altitudes of 
the respective places. 



METHODS OF ANOMALIES 173 

Two alternatives exist : 

1) There is a huge country with a high mountain range 
which is relatively not extensive in area, and the number 
of mountain stations is small (United States). 

2) There is a country the greatest portion of which is 
covered with high mountains (Switzerland). 

In problem 1, the actual temperatures at the elevated stations 
do not affect the general picture of isolines much, since the moun- 
tain stations are few. In the open, hilly regions, the representa- 
tion of the temperature distribution differs but little from the 
reduced isotherms, because of the small differences in height. 

In problem 2, the actual isotherms could be drawn only if the 
density of the network were exceedingly great. Otherwise the 
curves would cross crests and abysses, valleys and ridges, regard- 
less of the variations of temperature. Yet just these variations 
should be represented. 

A clearly legible number, giving the actual average temperature, 
beside each place-name (similar to the method of indicating the 
altitude on orographic maps), would be a much better expedient 
than lines, the trend of which is by no means certain and cannot be 
guaranteed, since the number of stations is never sufficient, owing 
to the great " relief energy/' 

It is generally recommended that numbers^characterizing the 
represented elements be added beside the station names (or station 
symbols) on maps of isolines, in any case. Only then is the reader 
able to control the trend of the lines and to change it according 
to his individual point of view. 

XIV. 5. STANDARD CURVES. ANOMALIES. ISANOMALS 

As already discussed, the influence of altitude upon the geo- 
graphical distribution of temperature should be eliminated by a 
sort of projection called reduction to a fixed level. Another attempt 
in this direction, utilizing the actual temperatures, has been made. 
It has been shown in the foregoing that neither the one nor the 
other method yields satisfactory results. 

The discussion in Chapter IV. 4, about "curve fitting/' con- 
cluded that the graphical Cartesian representation of two variables, 
an independent (e.g., altitude) and a dependent (e.g., temperature), 
is important for climatological purposes. 



174 METHODS IN CLIMATOLOGY 

These dot charts are interpolated either graphically or by 
means of analytical equations. Both methods permit the determi- 
nation of the values of the dependent variant for every value of 
the independent. Thus convenient tables can be constructed from 
which the values of the dependent variant can be taken for equi- 
distant values of the independent variant. A clipping from a 
table representing the average temperature-height curve above the 
bottom of a mountain valley was given in Table 16, as well as 
the corresponding graph Fig. 14, which can be called the standard 
curve. The numbers in the table represent the standard distribu- 
tion of the element with respect to the region in question. 

On the other hand, the actual value of the element at the place 
is known, as also its elevation. The difference, viz., actual value 
of the element minus standard value, is called anomaly. 

These anomalies are independent of altitude in the present case, 
because this is fully considered by means of the standard curve. 
Lines connecting places of equal anomaly are called isanomals. 

XIV. 6. APPLICATIONS OF THE METHOD 

OF ANOMALIES AND ISANOMALS 

* 

XIV. 6. a. Vegetative Period and Altitude 

The length of the vegetative period in Switzerland may serve as 
the first example. 3 

As was said previously, the duration (number of days) of a 
temperature above 43 F, derived from an average annual course of 
temperature, is called the vegetative period. 

These periods have been calculated by a simple interpolation 
formula (see VI. 2) for 168 stations in Switzerland. Their altitudes 
vary between about 800 and 8200 feet. The entire country, with 
mountains up to nearly 16,000 feet, has an east-west extension of 
about 186 miles, and a north-south extension of about 124 miles. 
This is, indeed, a mountainous country divided into a northern 
and a southern part by the crest-line of the Central Alps. 

From all the 168 stations together, suitably grouped according 
to elevation, the following analytical equation for the relation be- 
tween the duration in days (d) of the vegetative period and the 

3 V. Conrad, " Isanomalen der Andauer einer bestimmt vorgegebenen Tempera tur," 
Geografiska Annaler, 1929, p. 299. 

J. Maurer (Pas Klima der Schwciz, Frauenfeld, 1909, p. 68 ff.) gave anomalies and 
isanomals of annual average temperatures in Switzerland. 



METHODS OF ANOMALIES 175 

altitude in meters (ft) is obtained : 

d = 268 - 0.07 h 

From this standard equation, the standard values are derived for 
altitudes for every 200 meters (Table 44). 

TABLE 44. STANDARD VALUES (DAYS) OF THE DURATION OF 
THE VEGETATIVE PERIOD IN SWITZERLAND 



Height 
(meters) 


Duration 
(days) 


Height 
(meters) 


Duration 
(days) 


200 


254 


1200 


.. . . 184 


400 


240 


1400 


170 


600 


226 


1600 


156 


800 


212 


1800 


142 


1000 


198 


2000 


128 











The vegetative periods of two places, one north, the other south 
of the Alps, are compared in Table 45. 

TABLE 45. ANOMALIES OF THE DURATION OF THE VEGETATIVE PERIOD AT Two PLACES 
IN SWITZERLAND: Z NORTH OF THE ALPS; M SOUTH OF THE ALPS 



Height 
m 


Duration in days 
obs. standard 


Anomaly 
obs. standard 
days 


Z 355 


221 
269 


243 
243 


-22 
+26 


M . . 355 





The altitudes are by chance identical, so the two places have 
the same standard duration of 243 days. The actual duration has 
a deficit of minus 22 days north of the Alps; and a surplus of + 26 
days south of the Alps. The numbers (Table 45) are really illus- 
trative of the climate in these two zones. Everybody who crosses 
the Alps, especially in the late winter, is surprised by the climatic 
difference between north and south. He leaves the northern region 
with its ground everywhere covered with snow; but when he has 
passed through the Gotthard tunnel for instance, he arrives at the 
famous Lake of Lugano, where the trees are in full blossom. 

The anomalies mentioned above give an exact picture of a 
sharp climatic divide between the places Z, north of the crest, and 
M, south of the crest at 335 m (1160 feet). 

A place, Ri f 400 feet higher than M, has +25 days; and Br, 
1400 ft higher, has an anomaly of +30 days. The three places 
have similar exposure towards the south, and are well protected 



176 



METHODS IN CLIMATOLOGY 



against invasions of polar air. The differences are surprisingly 
small, in spite of the rather great differences in altitude. The 
examples show that apparently the influence of altitude upon the 
anomalies is practically eliminated. So it is possible to draw isa- 
nomals even in a region of great " relief -energy. 1 ' The anomalies 
combined with a standard element-height curve enable one to calcu- 
late the actual values for places where observations are available ; 
and, secondly, to interpolate with relatively small errors anomalies 
and actual values for any point. It is beyond the compass of this 
book to discuss the interesting picture resulting from application of 
the method of anomalies and isonomals. One example is shown in 
Figure 30, however: the isanomals of the duration of the vegetative 
period in Switzerland. In spite of the great differences in altitude, 




FIG. 30. Isanomals of the duration of the vegetative period in Switzerland. 

(After V. Conrad) 

on the southern slopes, a large region of the Alps is characterized 
by positive signs (a vegetative period longer than the standard 
duration) ; and north of the Alps, a zone of negative signs is seen. 
Not altitude but climatic exposure is represented. 

There is no doubt that the method of isanomals is, up to the 
present, the only method which eliminates the influence of altitude 
to a high degree, and is, thus, the unique method of graphically 
representing climatic conditions in a really mountainous region. 



METHODS OF ANOMALIES 



177 



It goes without saying that this method is practicable for any 
variable climatic element. 



XIV. 6. b. Temperature and Geographical Latitude 

The best known standard equation which correlates the de- 
pendent variable, the temperature (/) with the independent vari- 

TABLE 46. STANDARD DISTRIBUTION OF TEMPERATURE WITH GEOGRAPHICAL LATITUDE 
(After W. Meinardus and Hann-Stiring [5th Ed.]; Converted into F) 



* 


January 


April 


July 


October 


Year 


N90 


-42 


-18 


30 


-11 


-7.6 


85 


-36.6 


-15.7 


32.5 


- 8.0 


-6.2 


80 


-26.0 


- 8.9 


35.6 


- 2.4 


1.0 


75 


-20.2 


- 2.2 


38.1 


6.8 


5.5 


70 


-15.3 


6.8 


45.1 


15.3 


12.7 


65 


- 9.4 


18.9 


54.3 


24.6 


21.6 


60 


3.0 


27.0 


57.4 


32.5 


30.0 


55 


12.4 


35.2 


60.3 


37.2 


36.1 


50 


19.2 


41.4 


64.6 


44.4 


42.4 


45 


28.9 


50.7 


69.6 


52.7 


49.6 


40 


41.0 


55.6 


75.2 


60.3 


57.4 


35 


49.3 


62.6 


78.4 


66.0 


63.0 


30 


58.1 


68.2 


81.1 


71.2 


68.5 


25 


65.7 


73.8 


82.2 


76.3 


74.5 


20 


71.2 


77.4 


82.4 


79.5 


77.5 


15 


75.2 


80.1 


82.2 


80.6 


79.3 


10 


78.4 


81.0 


80.4 


80.4 


80.1 


5 


79.3 


80.2 


79.2 


79.3 


79.5 



Eq. 

S 5 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 
60 
65 
70 
75 
80 
85 
90 



79.5 

79.5 
79.3 
78.6 
77.7 
75.7 
71.4 
65.7 
60.1 
54.1 
46.6 
41.0 
35.8 
30.7 
25.7 
19.8 
12.6 
8.6 
7.7 



79.9 

79'7 

78.6 

77.4 

75.2 

71.2 

65.7 

59.4 

54.5 

46.4 

43.3 

35.2 

27.5 

19.0 

7.5 

- 6.7 

-19.8 

-28.7 

-33 



78.1 

76.8 

75.0 

72.1 

68.0 

63.5 

58.5 

53.2 

48.2 

43.2 

37.9 

27.7 

15.6 

3.0 

- 9.4 

-23.4 

-39.1 

-49.9 

-54 



79.7 

78.8 
78.3 
75.9 
73.0 
69.1 
64.4 
59.5 
53.1 
46.4 
41.7 
33.4 
24.8 
16.2 
6.1 

- 7.1 
-22.0 
-28.5 
-31 



79.2 

78.4 

77.5 

75.9 

73.2 

69.6 

65.1 

59.0 

53.4 

47.8 

42.4 

34.3 

25.9 

17.2 

7.5 

- 4.4 

-16.6 

-24.5 

-27.6 



178 METHODS IN CLIMATOLOGY 

able, the geographical latitude (<) is that of J. D. Forbes: 4 
t = - 17.8 + 44.9 cos 2 (0 - 630')C. 

This equation is derived from observations only of the northern 
hemisphere, but is valid, despite this fact, approximately from 
40S to 60N. 

Such an analytical equation is needed for theoretical investiga- 
tions. For practical purposes, such as calculating anomalies, 
Table 46 is preferable to an equation. 

The values in the table 6 are probably the best available. 
Particularly, the formerly dubious temperatures of the high south- 
ern latitudes are replaced by those of W. Meinardus, who derived 
them from the observations of the most recent expeditions. The 
series of temperatures of January, July, and the year in the table 
were revised by Meinardus himself in 1936. The centigrade 
temperatures of the original table appear in this book converted 
into Fahrenheit degrees for the first time. 

Naturally, the series of Table 46 should be interpolated for each 
degree, if used for calculating the anomalies of a great number of 
places. 

In view of the great importance of the method of anomalies, 
Figure 31 has been inserted, ill order to show how much a repre- 
sentation like this contributes to the understanding of the effect 
of ocean currents and continentality upon the temperature at the 
earth's surface. 

XIV. 6. c. Pleions, Meions 

Another application of anomalies and isanomals has been given 
by H. Arctowski in a series of papers. 6 

For a portion of the earth's surface, annual averages of an 
element are known for a normal period. Anomalies are computed 
for each year of the period in question and represented by maps of 
isanomals. Regions encircled by a high positive isanomal are 
called pleions those characterized by high negative anomalies are 
named antipleions or melons. The method is applicable to every 
variable element. If temperature is dealt with, the terms thermo- 

4 James D. Forbes, " Inquiries about Terrestrial Temperature," Transactions of the 
Royal Society of Edinburgh, XXII (1859), pt. I, p. 75 ff. 

6 See Hann-Siiring, Lehrbuch der Meteorologie, 5th ed., and V. Conrad, " Die klimato- 
logischen Elemente . . .," Koppen-Geiger, Handbuch der Klimatologie, IB, 123. 

6 E.g., UEnchainement des variations climatiques (Bruxelles: Socidt^ Beige d'As- 
tronomie, 1909); "Zur Dynamik der Klimaanderungen," Met. Zeit. 1914, pp. 417-426; 
Communications de I'Institut de Geophysique et de Meteorologie de V Universite de Lw6w. 
See also the valuable and critical review by Alfred Wegener, in Gerland's Beitrdge zur 
Geophysik, vol. X, "Besprechungen," pp. 298-299. 



METHODS OF ANOMALIES 



179 




180 METHODS IN CLIMATOLOGY 

pleion and thermomeion are also used. An interesting attempt has 
been made to study the shifting of the pleions and meions from 
year to year. These efforts might be successful for the purposes 
of dynamic climatology. (XVIII. 6.) 

XIV. 6. d. Snow-Cover and Altitude 

The duration of snow-cover has been represented by the method 
of isanomals for the eastern Alps and illustrates the influence 
of a high mountain range extending east and west, upon snow 
conditions. 7 

XIV. 6. e. Duration of Sunshine vs. Elevation 

A study was made of the distribution of bright sunshine dura- 
tion in the eastern Alps, by the method of isanomals. 8 The map 
(Fig. 32) shows that the positive anomalies are crowded on the 
southern slopes of the Alps, and the negative on the northern. 

Thus, the method of isanomals again gives the trend of the 
effective climatic divide between the northern and southern slopes 
of the Alps. 

This result is of great consequence for the appreciation of the 
method of isanomals, and runs parallel with the method of correla- 
tion, 

XIV. 6. f. Anomalies of Precipitation 

The knowledge of the geographical distribution of rainfall is 
of the greatest importance for agriculture, hydrodynamics, archi- 
tecture, etc. Conversely, it would be difficult to find a branch of 
human activity which is not interested directly or indirectly in 
the amounts of precipitation which one may expect everywhere on 
the earth's surface. So it is no wonder that innumerable maps of 
the distribution of precipitation appear in books, encyclopaedias, 
journals, papers, and pamphlets. The difficulties in drawing such 
maps have not been overcome. 

As long as the region for which the precipitation is to be shown 
is flat, like a table, the use of isohyets, i.e., lines of equal precipita- 
tion, is more or less satisfactory. If the country is hilly or even 
mountainous, then difficulties set in, as in the case of temperature. 

7 V. Conrad, M. Winkler, "Beitrage zur Kenntnis der Schneedeckenverhclltnisse 
in den oesterreichischen Alpenlandern," Gerland's Beitrage zur Geophysik, XXXIV 
(1931), 473. 

*V. Conrad, Anomalien und Isanomalen der Sannenscheindauer in den oester- 
reichischen Alpen (Wien, 1938). 



METHODS OF ANOMALIES 



181 




O 

U 



o> 



Si 

G 



en 

Q, 



C 

s 

en 
<rt 

CD 

<u 

-5 



S 






, 'a 
L 8. 



O 



o 

V 



O 

I 



182 METHODS IN CLIMATOLOGY 

The comparison of like with like is not valid. The difficulties are 
even much greater than with temperature. With this last, there 
is great trouble with inversions in winter in the mountain valleys. 
Regardless of this real obstacle, the normal lapse rate (of tempera- 
ture) does not vary much over the surface of the earth, if great 
regions are considered and details excluded. 

For example: The lapse rate in Peru, 16S, is 0.61 C/100 m; 
in Ceylon, 7N, 0.64; on the slopes of Mount Etna (Sicily), 38N, 
0.64; on Ben Nevis (Scotland), 57N, 0.67 C/100 m. 

For precipitation, the conditions are of an entirely different 
nature. For example: for Christmas Island (10S, Indian Ocean, 
190 miles south of Java) the author calculated the increase of rain- 
fall to be 140 mm /1 00 m. On the slopes of the volcano Tengger 
(Java), C. Braak found 97 mm/100 m, and that a decrease, between 
600 m and 1700 m altitude. On the slopes of Mount Idgen 
(Java), an increase of 260 mm/100 m was stated. 

In a small valley a few miles long in the European Alps, the 
increase of precipitation, even with differences in height up to 
more than 7000 feet, varies from 14 mm/100 m to 163 mm/100 m 
(0.17 in./lOO feet to 1.94 in./lOO feet). In every mountain range 
of temperate latitudes a decrease of precipitation with increasing 
altitude occurs. 

Another striking example of these conditions is presented by 
O. Liitschg. 9 In the small region of the famous Mont Cervin 
(Matterhorn), the following variations of precipitation with height 
are observed : 

Variation of 
Precipitation 
with Height 
Pair of Stations mm/ 100 m 

Visp Zermatt increase 9.3 

Grachen Zermatt increase 912.0 

Visp Grachen decrease 5.8 

Visp Saas Fee increase 20.8 

Saas Tamatten Saas Fee invariance 

In the Rhone valley (Wallis), the conditions mentioned are 
even more exaggerated : 

9 Otto Ltitschg, " Ueber Niederschlag und Abfluss im Hochgebirge," Verd/entlichung. 
der hydrologischen Abteilung der Schweizerischen Meteor ologischen Zentralanstalt in Zurich 
(Ztirich, 1926), p. 137). 



METHODS OF ANOMALIES 183 

Variation of 
Precipitation 
with Height 
Pair of Stations mm /1 00 m 

Martigny-Ville Sion increase 347 

Sion Sierre decrease 200 

Sierre Visp increase 84 

Reckingen Oberwald increase 2252 

Oberwald Gletsch increase 46 

In view of these conditions no reduction to a fixed level is 
possible. Therefore, the usual rain map is the representation of 
actual isohyets. 

What was said about actual isotherms is true also for actual 
isohyets, owing especially to the great irregularity of the phe- 
nomenon in regard to time and locality. The difficulties become 
very complex, but mostly in hilly and mountainous regions. Here 
again, the method of the isanomals is the only one that is able to 
eliminate the influence of height and to represent the true features 
of rainfall distribution. This method determines, in this case, too, 
the position of the climatic divide, even if it is shifted toward the 
lee side. The reason for this phenomenon belongs to the realm of 
physical climatology. 

XIV. 7. PRECIPITATION PROFILES 

In view of the great difficulties with isohyets, profiles of precipi- 
tation drawn through the mountain ranges are, at least theoret- 
ically, a good expedient; theoretically, since the profiles can be 
drawn successfully only if the observations of a sufficient number 
of suitably located stations are available. 

The principal requirement is that there be stations at or close 
to the points of the profile extremes. If this condition is not 
satisfied, the drawing of the profile is nothing but a guess, and the 
profile itself represents fancy more than truth. The stations 
should not be too scattered on either side of the location of the 
profile. 

The network of stations up to the crests and summits of the 
high mountains is very dense in Switzerland, so an example from 
the western Alps may be useful. This is reproduced in Figure 33, 
where Liitschg 10 gives a profile through the Bernese Alps from 
NW to SE. 

10 0. Ltttschg, Ueber Niederschlag und Abfiuss im Hochgebirge (Zurich, 1926), p. 138. 



184 



METHODS IN CLIMATOLOGY 



This profile may serve in some points as a pattern. The 
vertical scale is not so exaggerated as is often the case in such repre- 
sentations. It is 10.5 times greater than the horizontal. The 
length of the profile is 278 miles, with 27 stations. On the average, 



Bern- JungFraujoch-Qeisspfadsee- Domodossola - Pavia - 



* n Ma ^ pHno 




FIG. 33. Precipitation profile across the Bernese Alps from NW to SE. 

(After O. Lutschg) 

there is one station every ten miles. This ratio does not give the 
right picture, because, deliberately, the stations are not evenly 
distributed over the profile. The stations are very dense within 
the region with greatest differences in height. There, 10 stations 
are located within 37 miles. In the plain of the Po (Italy), two 
stations are shown within 60 mil^s. 

It is usual to blacken the mountain profile. In the figure, it is 
slightly hatched so that certain levels can be indicated by hori- 
zontal lines. This gives valuable information. 

The mountain and the precipitation profiles are drawn sepa- 
rately: the two scales do not overlap, but are arranged one above 
the other, with homologous points on the same vertical. 

At the southern side (right), the profile is bifurcated, showing 
the amount of rain for the plain on the one hand, and for the hills 
(about 3000 ft) on the other. 

The clear marks and lines showing the vertical and the hori- 
zontal scale, should not be overlooked. 

This representation is well-planned, indeed. Yet one un- 
familiar with Swiss orography, looking at the graph, might not 
understand why the amounts increase so much between the 100 
km (see figure 33) and the 150 km point. Here, the profile is 
influenced by the contour features on either side. 

Nevertheless, profiles based on sufficient and exact data are an 
excellent and necessary supplement to the maps of actual isohyets, 
never satisfactory in mountainous countries. 

Figure 34 exhibits a profile of snowfall across New England 
from the New Hampshire coast to Williamstown, Mass. C. F. 



METHODS OF ANOMALIES 



185 




FIG. 34. Profile of snowfall across New England from the New Hampshire 
coast to Williamstown, Mass. (After C. F. Brooks) 

Brooks's graph n is, on the whole, in good agreement with the 
foregoing suggestions. Representation of the hypothetical por- 
tion of the curve by a broken line might be generally adopted. 
An arrow indicates the direction of the snow bearing winds. 

XIV. 8. VARIABILITY OF PRECIPITATION AND 
AMOUNTS OF PRECIPITATION 

The independent variant is the amount of precipitation; the 
dependent is the variability. This combination shows how adapt- 
able the method of anomalies and isanomalies is. The problem 
was discussed in III. 8. 

"Charles F. Brooks, "The Snowfall of the Eastern United States," M.W.R., 
January 1915, vol. 43, p. 9. 



CHAPTER XV 
WIND 

XV. 1. WIND ROSES 

f "^HE WIND ROSE is defined as "a diagram showing for a 
I definite locality or district, and usually for a* more or less 
JL extended period, the proportion of winds blowing from 
each of the leading points of the compass." l In the simplest 
form, eight directions are used, and relative frequencies (per cent) 
should be preferred. The frequency of calms is indicated by a 
circle in the center of the rose. The radius is proportional to the 
frequency; or, as on United States pilot charts, a number in the 
circle shows the per cent of calms. 2 The method is too well known 
to require further explanation. 

Sir Napier Shaw constructed a composite wind rose or wind 
"star" in the utmost detail. 3 This is reproduced in Figure 35. 
It shows the annual course of the ffequency of three classes of wind 
velocity for 16 directions. 

The three steps of velocity are : 

1) light winds 

2) moderate or strong winds 

3) gales 

Light winds (Beaufort 1 to 3), are represented in the rose by a single 
line. The second group includes the Beaufort degrees 4 to 7, 
and is indicated by a double open line. At their ends are small 
blackened sections which represent the number of gales of force 
8 and more. Frequencies are given in per cent. "The indenta- 
tion of the base line allows us to distinguish the information for 
the summer half, April to September, for the northern hemisphere, 
from the two winter quarters, January to March on the left for a 
spectator looking from the centre and October to December on the 
right. Small letters A and 5, hardly visible in the reproduction 
without a glass, mark the ends of the salient." Thus, one "star" 

1 Meteorological Glossary (3d ed., London, 1939). 

1 For a good example of wind roses and their applications see Glenn T. Trewartha, 
An Introduction to Weather and Climate (2d ed., New York, 1943), p. 112. 
8 Sir Napier Shaw, The Drama of Weather (Cambridge, 1933), p. 165. 

186 



WIND 



187 



replaces twelve normal wind roses. Drawing one star may be more 
troublesome than drawing twelve simple roses, but much printing 
space is saved. Wind roses are often used to indicate the geo- 
graphical distribution of wind conditions, and are legible even on a 
rather small-scale map; but this is not true for the "stars." 




FIG. 35. The wind "star" (composite wind rose). 
(After Sir Napier Shaw) 



For climatological purposes, the data used for wind roses are 
better given in numerical tables, which take into consideration also 
the frequency of calms. Here, too, percentages are preferable to 
absolute frequencies. 4 The horizontal entry of these tables of 
distribution of wind directions has headings for the eight directions 
and the calm ; the vertical is used for the months. Such statistics 
are indispensable. 

XV. La. Wind Roses for Different Elements 

Wind roses for different elements are based on the principle 
mentioned, and are illustrated by the example in Table 47. 6 The 

4 See models in R. DeC. Ward and C. F. Brooks, "Climates of North America," 
K6ppen-Geiger, Handbuch der Klimatologie, pt. J, pp. 234 ff. 

After A. Wijkander, "Observations mt&>rologiques de 1'expddition arctique 
Su&ioise 1872-73," Kongl. Svenska Vctenskaps-Akad. Handlinger Bandet 12 t no. 7 
(Stockholm, 1875). 



188 METHODS IN CLIMATOLOGY 

TABLE 47. WIND ROSES FOR DIFFERENT ELEMENTS OBSERVED AT MOSSEL BAY, 

SPITSBERGEN (80N, 15E), 1872/73. Deviations for Temperature (A*C) and 

Cloudiness (AC%), and Per Cent of Probability of Precipitation (P-Pro %) 

(After A. Wijkander) 

Wind 
direction A*C AC% P-Pro % 



N 


-1.8 


+ 9 


33 


NE 


-2.4 


4 


18 


E 


2.3 


17 


8 


SE 


+0.5 


-13 


5 


s 


+4.9 


- 6 


6 


sw 


+3.8 


+ 3 


19 


w 


-0.4 


+ 14 


40 


NW 


-2.0 


+ 16 


38 



Calms -1.1 -10 11 

NOTE: To avoid misunderstanding: The " P-Pro " percentage relates to single wind 
directions; e.g., there are 33 rainy days out of 100 days with N-wind, and only 5 rainy 
days out of 100 with SE-wind. The representation clearly shows that a strong tendency 
toward precipitation exists with northerly and westerly winds, and a small one with 
southerly and easterly winds. 

numbers in the columns A/C and AC% denote average deviations 
for the eight directions and for caliji. " P-Pro " means the average 
probabilities of precipitation. The method combines the immedi- 
ately observed surface winds with the synchronous values of the 
different elements. 

Wind roses are frequently a good climatic characterization, 
but do not contribute much to air-mass analysis. The correlation 
between the momentary surface wind and the life history of an air 
mass is not well defined. 

XV. 2. MAPS OF STREAMLINES 

For a cartographical representation, maps of streamlines seem 
to offer the best climatological information. 

Streamlines exhibit the state of air motion at a particular 
moment, and must be carefully distinguished from trajectories or 
actual paths of an air particle. Streamlines reproduce the synoptic 
picture at a given moment, trajectories the chronological story of 
an air particle. Maps of streamlines should be accompanied by 
maps of the synchronous distribution of air pressure. 

The technique of drawing streamlines includes the considera- 
tion that the resultant wind direction at a place should be the 
tangent to the streamline. Besides, physical considerations con- 



WIND 



189 



earning the field of pressure and temperature are of primary 
interest in drawing streamlines. 

As was stated in Chapter VIII, the observations and publica- 
tions include either (a) direction and velocity, or (b) only a fre- 
quency distribution. In the first case, the resultant wind velocity 
and steadiness (an important characteristic of wind conditions) can 
be computed for every station. 

Climatological streamlines represent, of course, average condi- 
tions. As far as the wind is concerned, a climatological streamline 
does not mean a reality like a meteorological streamline : the distinc- 
tion is the same as between an arithmetical mean and an actual 
value. The arithmetical mean becomes the more characteristic the 
smaller the variability. At a tropical place with a very small tem- 
perature variability the monthly mean of temperature is no longer 
a fictitious number. In wind conditions " steadiness" replaces 
" variability." The higher the steadiness, the more the average 
streamline approximates actuality. Therefore, it is of great in- 
terest to add the representation of steadiness to that of velocity 
and direction on a climatological map of streamlines. 




FIG. 36. Streamlines over the Balkans; average for July 1917. 
(After E. Kuhlbrodt) 

As an example, average streamlines are given for the eastern 
Balkans, according to observations made in July 1917. The special 
method applied to the construction of this map (Fig. 36), has been 
elaborated by Alfred Wegener, and fulfills apparently all require- 



190 



METHODS IN CLIMATOLOGY 



ments. The trend of the streamlines in Figure 36 is interesting, 
in that the graph clearly shows how the general west-east streaming 
of the air over the higher latitudes of Europe is deflected to winds 



A 



/'v 



V 



f 

S 

/' 

^' 



/ 




--X-~-j f 
x \ s i 

... .-JTO ^ - 

\_J^\ 




JANUARY 
STREAMLINES 

NEW ENGLAND 



FIG. 37a. Streamlines of New England in January. (After V. Conrad) 

with a northern component; this is more and more strengthened 
towards south and east. So the map constitutes a representation 
of the "Etesians." The feathers, marked directly on the stream- 
lines, indicate the velocity of the wind in relative measure. In the 



WIND 



191 



present map, half feathers are equivalent to the weakest winds, 
and three full feathers indicate a very strong average velocity. 
(See the region off the coast of Asia Minor.) 




JULY 
STREAMLINES 

NEW ENGLAND 



FIG. 37b. Streamlines of New England in July. (After V. Conrad) 

The steadiness is expressed in the following way: 

1) If the streamlines are dotted, the steadiness is between 
0% and 33% 

2) Broken streamlines mean a steadiness of 33% to 66% 

3) Full streamlines indicate a steadiness greater than 66% 



192 METHODS IN CLIMATOLOGY 

From the present representation, one can see that the northerly 
winds of the Aegean Sea are not only the strongest winds but also 
those with greatest steadiness. 

XV. 2. a. Streamlines Based Only upon Frequencies of Direction 

If records of velocities are lacking, average resultant wind 
directions may nevertheless be computed, as was proved earlier 
(see VIII. 2). Then, of course, only streamlines can be con- 
structed; naturally no velocities, no steadiness can be indicated. 
Even a plain streamline picture is climatically instructive, as is 
shown in Figures 37 a and b. At first glance, the reader sees the 
relatively slight shifting, even if climatically decisive, of the average 
wind from winter to summer. The continental features of winter 
on the New England coast clearly appear, and the warm, humid 
air masses which overflow the country in summer are indicated. 



CHAPTER XVI 
AIR MASS CLIMATOLOGY 

AERAGE STREAMLINES (mentioned in the foregoing chapter) 
are closely connected with the concept of air masses, so 
that this branch of climatology, rich in prospect, may be 
discussed here. The methods used in these investigations are 
similar to the usual statistical procedures. They start from the 
frequencies of different types of air masses for certain parts of the 
year. An air-mass calendar taken from the daily weather maps is 
the foundation. Relative frequencies should be preferred. 

One of the results is also of special methodological interest: 
Air masses can be divided into two main groups: continental (c), 
and maritime (m). The frequency-ratio c/m is obviously a good 
characteristic of the climate, and perhaps a better index of con- 
tinentality than the factors of continentality (discussed later on). 

Table 48 offers two examples of the seasonal ratios c/m, one 
for western Europe, 1 the other for the eastern United States. 2 

TABLE 48. DEGREE OF CONTINENTALITY DERIVED FROM FREQUENCIES OF 

CONTINENTAL (c) AND MARITIME (m) AIR MASSES (FOR PENNSYLVANIA 

AND FOR WESTERN EUROPE) 





Winter 


Spring 


Summer 


Fall 






PENNSYLVANIA 






c/m 


1.65 


1.08 


1.02 


1.07 






WESTERN EUROPE 






c/m 


0.58 


0.40 


0.18 


0.25 



The results of Table 48 do not need further explanation. In 
the eastern United States, especially in winter, the continental 

1 E. Dinies, Luftkorper Klimatologie (Frankfurt a.M., 1932) (air mass system pro- 
posed by F. Linke). 

'H. Landsberg, "Air Mass Climatology for Central Pennsylvania," Gerland's 
Bcitrage zur Geophysik, vol. 51 (1937) p. 263 (H. C. Willett's classification). For further 
information see Jerome Namias, Air Mass and Isentropic Analysis (5th ed., with con- 
tributions by T. Bergeron, B. Haurwitz, G. Miller, A. K. Showalter, R. G. Stone, and 
H. C. Willett). In this connection, chapter n on "Classification of Air Masses," with 
list on p. 76, is of particular interest. See, for an informative survey, chapter iv, 
"Air Masses and Fronts," in Glenn T. Trewartha, An Introduction to Weather and Climate 
(2d ed., New York, 1943), pp. 190 ff. 

193 



194 METHODS IN CLIMATOLOGY 

component predominates; in western Europe, the maritime. The 
first result is confirmed by the average streamlines in winter over 
New England (Fig. 37). The contrast between the two climates 
is well described by the ratio c/m. 

The great merit of the air mass index of continentality is that 
it needs no special assumption or hypothesis. Thus, it yields 
incontestable results. The disadvantage is that the basic air 
mass calendar entails much work and critical study. 



CHAPTER XVII 

NUMERICAL CHARACTERIZATION OF DIFFERENT 
CLIMATIC FEATURES 

XVII. 1. CONTINENTALITY 

f "^HE INCREASE of the annual range of temperature inland is 
I perhaps the most striking effect of the continental surface 
J- on climate. Conversely, the annual range (-4) is taken as 
a measure of the climatic factor, called continentality. Because a 
general physical correlation exists between annual range and geo- 
graphical latitude, the range has to be reduced to equality for all 
latitudes ; thus the expression 

A 

sin 
is the measure sought for. 

The formula, now frequently used, reads: * 

C = L7 . X A - 20.4 
sin 

where C is the coefficient of continentality in per cent, A the an- 
nual range of temperature in centigrade degrees, and < the geo- 
graphical latitude. 
Example : 

1) Jakutsk (62.0N, 129.7E, 320 ft.) A = 61.6 C 

2) Thorshavn (62.0N, 6.8E, 83 ft.) A = 7.6 C 

Therefore 

1.7 X61.6 _ 

Cjak - ~ 0.88 ' = 

Cxhor.. = L7 * 8 7 ' 6 - 20.4 = - 6% 

The second example shows that the constants have to be some- 
what changed, since a " negative continentality" has no physical 
meaning. If the continentality is zero, the climate is no longer 
influenced by continental surfaces in any way. Consequently, it 
is always better to speak of continentality than of "oceanity." 

1 Wl. Gorczynski, "Sur le calcul du degr du continentalisme et son application dans 
la climatologie," Geografisker Annaler, 1920, p. 324. 

195 



o 



196 METHODS IN CLIMATOLOGY 

From the methodological point of view, the fact of a negative 
coefficient is not so important and easily corrected ; the later com- 
ments by Gorczynski himself and by O. V. Johansson are valuable 
in this respect. The really weak point, however, has (to the best 
of the writer's knowledge) never been emphasized, simple as it 
may be. 

The coefficient of continentality C becomes infinite (C = <*>), 
or practically infinite, in the interior tropical belt, if sin <f> 0. 
Therefore, all these formulae, going back to W. Zenker's investiga- 
tions, are no longer valid for very low latitudes. This point is 
fundamental in regard to method, and shows that one must always 
take into account the boundary conditions of the formulae used. 

On the other hand, it should not be forgotten that the formula 
gives a comparably numerical representation of the continentality, 
although for only a part of the earth's surface. 

Discussing a formula of continentality given by R. Spitaler 
and mapped by G. Swoboda, D. Brunt 2 arrives at an equation 
which shows a clearer physical significance than that mentioned. 
His formula reads : 

n e + 0.12 = A//130.61-AS 

where n e denotes the continentality -factor of a given place, A/ the 
annual range of temperature, and AS the annual range of the aver- 
age intensity of solar radiation in the latitude of the place in 
question. 

Then, "(n e + 0.12) measures the response in monthly mean 
temperature to a unit change in the mean intensity of solar 
radiation." 

We have only to add that D. Brunt alludes to the point that in 
the maps of Swoboda no isolines are drawn within the zone 20N 
to 20S, "on account of the uncertainty in the computation of 
n e for this region." This remark is in good agreement with the 
criticism above mentioned. 

XVII. 2. LIMITS BETWEEN TIMBER-FOREST, STEPPE, AND DESERT. 
EFFECTIVENESS OF PRECIPITATION 

Supan and Koppen derive numerical limits of their climatic 
provinces from average data of temperature and precipitation. 
For example : 

1) The coldest month in the "tropical (A) -Climate" is 
warmer than 64F. 

David Brunt, "Climatic Continentality and Oceanity," Geographical Journal, 
London, vol. LXIV (1924), p. 43. 



CHARACTERIZATION OF CLIMATIC FEATURES 197 

2) The highest average monthly temperature within the 
41 regions of snow and frost " (EF-Climates) is below 
32F., etc. 

Supan had already defined the tropical zone by the annual iso- 
therm of 68 F. This definition marked a great advance in this 
subject. That was also Koppen's opinion, although his numerical 
limits are incomparably more practical. 

Great difficulties arise if the effect of precipitation has to be 
estimated. Chief of these is the problem of the limit between 
steppe and timber-forest on the one hand and steppe and desert 
on the other. 

The consequences of a given amount of rain are dependent on 
temperature, owing to the effect of evaporation. It is impossible, 
either theoretically or practically, to formulate an equation which 
would take account of all effective elements and factors, such as 
radiation, wind, soil, continentality, humidity, elevation, etc. 
Therefore, Koppen had to be satisfied with rough empirical ap- 
proximations. 3 He himself frequently changed these formulas, so 
that one example may suffice. Often a slight modification of 
Koppen's rules helps if a special region has to be described. 

The effectiveness of precipitation is greater at low tempera- 
tures. Consequently, for instance, one must discriminate between 
steppes with precipitation in cool winter and steppes with precipita- 
tion in hot summer. 

The limit between a forest climate and a steppe climate is 
indicated by the following relations between temperature, t (C), 
and precipitation, r (cm) : 

a) rainy period definitely in winter: r = 2t 

b) rainy period definitely in summer: r = 2t + 28 

c) no definite rainy period: r = 2t + 14 

The limit between steppe and desert can be estimated to lie 
where the rainfall is one-half of the amount stipulated above, in 
each case : 

a) winter rain r = / 

b) summer rain r = / + 14 

c) transition cases r = / + 7 

The following example may be instructive : 

1 For further details, see W. Ktippen, Grundriss der Klimatologie (2d ed., Leipzig, 
1931), and Kflppen-Geiger, Handbuch der Klimatologie, vol. Ic (1936). 



198 METHODS IN CLIMATOLOGY 

1) Stalingrad (48.7N, 44.5 E, 138 ft):/ = 7.7C;r = 3 

2) Achtuba (48.3 N, 46.9E, 11 ft):/ = 7.7C;r = 2S cm 

3) Astrachan (46.4N, 48.0E, -30ft):/ = 9.4C;r = 15 cm 

Astrachan lies in the delta region of the Volga river, Achtuba is 
150 miles distant in the Volga basin, and Stalingrad lies another 
120 miles up the river. For the present purpose one can assume 
in rough approximation that the distribution of precipitation over 
the year is more or less uniform. Therefore the formulas (c) 
should be valid. For the temperature of Stalingrad and Achtuba 
we get the formula : 

steppe climate if r < (2t + 14) = 29 cm 

Therefore Stalingrad with 37 cm has a forest climate: Achtuba 
with 25 cm has a steppe climate, and the climate of Astrachan is 
that of the desert, because there is less than 16 cm rainfall. 

The boundary between forest and steppe climate runs between 
Stalingrad and Achtuba. The border line of the desert can be 
assumed to lie between Astrachan and Achtuba, but closest to 
Astrachan. An interpolation regarding the location of such a 
border line between two places should be made with the greatest 
caution. The reader may be reminded of coherent and non- 
coherent climatic regions. Every interpolation presumes con- 
tinuity of the variations. 4 

The index of precipitation effectiveness given by C. W. Thorn- 
thwaite 6 reads : 

(P \10/9 
r^To). 

The symbol / is called the precipitation-effectiveness index. P 
means the monthly precipitation in inches and T the average 
monthly temperature in F, while n indicates the number of the 
month in question. 

There is no doubt as to how to calculate this index, but it is a 
rather wearisome task. The reader who intends to use this 
formula finds some help from the nomogram given by Thorn- 
thwaite. 

4 Since the stations are generally far distant from one another, the interpolation 
of the values, critical for the location of the border lines between forest and steppe 
and between steppe and desert, is not an entirely reliable procedure. Therefore, 
botanical facts also ought to be considered in drawing these border lines. 

5 C, W. Thornthwaite, "The Climates of North America According to a New 
Classification," Geographical Review, October 1931; " The Climate of the Earth," ibid., 
July 1933; Atlas of Climatic Types in the United States 1900-1939 (Washington, 1941). 



PART IV 
THE CLIMATOGRAPHY 

CHAPTER XVIII 
ARRANGEMENT OF A CLIMATOGRAPHY 

IN THE EARLIER PARTS of this book, methods of examining 
observational data in a critical and quantitative way were 
explained. From the homogeneous and reduced series of 
elements, characteristics were derived by means of simple mathe- 
matical statistical methods. 

The idea which runs right through is to proceed from the 
qualitative to the quantitative, and to arrive at comparable and 
reproducible results. These are the methods of analysis. 

But a climatography should give more than this. It should 
offer a full picture of the climate in question. This aim can rarely 
be reached. Nevertheless, the content, outlines, and arrangement 
of a more or less complete climatography are here suggested. 

XVIII. 1. THE INTRODUCTION 

The Introduction should contain a short geographical descrip- 
tion of the region in question, illustrated by a clear, legible, oro- 
graphic map. At least the trends of rivers, streams, and lakes, 
as well as those of the mountain ranges, should be outlined. 

A second map showing the locations of the stations should not 
be omitted. The reader finds a pattern of the latter in each 
section of the Koppen-Geiger Handbuch der Klimatologie, e.g., 
in R. DeC. Ward and C. F. Brooks, "Climates of North America." 
But a net of parallels and meridians running right across the map 
should not be forgotten. An alphabetical list of stations is indis- 
pensable. The three geographical coordinates should be added to 
each station name, with a number referring to the map of stations. 

The second part of the introduction should give a survey of 
the material in connection with the map of stations. The kind of 
shelters and rain gauges and their exposure at the majority of 
stations should be briefly and clearly described. The history at 

199 



200 METHODS IN CLIMATOLOGY 

least of the " normal stations" should be given, pointing to chang- 
ings of exposure of the instruments, of methods of observation, of 
operators, etc. 

Smoothing of certain series correcting obviously false observa- 
tions and interpolating of missing values should be indicated. 

The series whose relative homogeneity has been examined 
should be given in extenso, as well as detailed results of these com- 
putations with an exact indication of the methods applied. Simi- 
larly, a clear report with all necessary numerical additions should 
be given regarding the reduction to the chosen " normal period." 
It does not suffice to make a general statement that the series are 
homogeneous and that they are reduced to a given period. Only 
from detailed numerical results can the reader judge how far the 
representation is reliable and useful. 

XVIII. 2. STATIC CLIMATOLOGY 

The static climatology deals with the average state of the 
atmosphere. It is more or less identical with climatological 
statistics. Therefore, the backbone of this part is represented by 
material which must be given in numerical tables and graphs. 

There are different kinds of tables. 

1) The climatic table contains average values for a certain 
place, derived from a certain period, and besides, extreme values. 
Samples of this kind of table are given in the appendix. The title 
of each climatic table must show, also, the name of place and 
country, and the three geographical coordinates. 

It often happens that it is impossible to give all the data for 
one and the same period. Then the period must be mentioned at 
the head of its column, which, in any case, has to contain the units 
of the respective element. (C, F, tercentesimal degrees or ab- 
solute degrees, "A," or "K" = Kelvin; inches or millimeters, 
millibars, millimeters of mercury or inches of mercury, meters, 
feet, etc.). 

2) The climatic table can also be replaced by tables each of 
which is related to a certain element (average temperature, cloudi- 
ness, etc.) and contains a list of stations and the averages of the 
element in question for the months and for the year. As a sample, 
again the tables in "The Climates of North America" (R. De- 
Courcy Ward and C. F. Brooks) can be recommended. Many 
authors prefer tables of this kind because they save space and 
printing costs. Climatic tables should be given, in any case, for 



ARRANGEMENT OF A CLIMATOGRAPHY 201 

the most representative places of the region. If a climatography 
of New Hampshire were composed, climatic tables would have to 
be given for Mount Washington, for a suitable station at its base, 
and for at least one station near the sea. 

3) For stations with more than ten years' operation, tables 
should be included which contain the monthly and annual means 
or sums, at least, for temperature, rainfall, and pressure, for all 
available months and years. Cloudiness should be added, if in 
any way possible. 1 

Whatever kind of climatic tables is considered, frequency dis- 
tributions (see II. 4 and ff.) of the various elements contribute 
essentially to the quantitative description of climate and can be 
advantageous in many even not specifically climatological prob- 
lems. Recently J. Namias (" Construction of 10000-Foot Pres- 
sure Charts Over Ocean Areas/' BAMS vol. 25, 1944, p. 177 ff.) 
has used frequency distributions of air temperature by 5 squares 
over the Atlantic Ocean for his forecast technique. 

XVIII. 3. MONOGRAPHS 

Monographs are of great value, inasmuch as they deal with 
long homogeneous series of different elements obtained at a place 
representative of a greater region. It is impossible to give specific 
instructions for composing such investigations and publications. 
The more details and statistics are given, the more useful the 
monograph is. The reader may be referred to some models of 
different kinds: 

1) Under the direction of Dr. J. B. Macelwane, S.J., there 
appeared Meteorological Means and Normals from Observations at 
Saint Louis University, ipu-ipJ5, by John J. Renk, S.J., two 
volumes which offered only numerical tables, extremely useful for 
other scholars. 

2) A. Wagner, in Der Jahrliche Gang der meteor ologischen Ele- 
ments in Wien (Wien, 1930), presents a full discussion and a great 
collection of statistical tables. Both monographs are restricted to 
the annual course of the elements. 

3) F. Steinhauser, in Die Meleorologie des Sonnblicks (Wien, 
1938), gives the pattern of a monograph which considers the annual 

1 For examples see R. DeC. Ward and C. F. Brooks, " Climates of North America/' 
in Kdppen-Geiger, Handbuch der Klimatologic, p. 266 ff, or U. S. Weather Bureau, 
41 Climatic Summary of the United States," e.g., W. M. Wilson, Section 82, South Central 
New York, p. 3 ff. 



202 METHODS IN CLIMATOLOGY 

course as well as the daily, in an exact way, by numerical data and 
their discussion. 

A different kind of monograph is that of O. L. Fassig, The 
Climate of Baltimore. It will be mentioned later (XVIII. 7.) 

XVIII. 4. RECORDS OF SELF-REGISTERING INSTRUMENTS 

If continuous records are at hand, the daily course of the re- 
spective elements should be shown. Often, two-hourly values are 
sufficient. In the first place, the daily variation . has to be re- 
produced numerically. Graphs are always welcome but do not 
replace numerical tables. From the tables the reader can make 
himself a correct graph. The converse is not true. 

From the continuous records periodic and aperiodic variations 
have to be derived. 

The periodic variation is the difference between the highest and 
the lowest value of the average daily course (perhaps for the 
period of a month) of the element (pressure, temperature, relative 
humidity, etc.). 

The aperiodic variation is obtained from the average daily 
extremes. In the case of temperature, it can be obtained not only 
from the records of the thermograph but also from the daily 
readings of the extreme thermometers. 

XVIII. 5. CHANGE OF TEMPERATURE AND 
PRECIPITATION WITH HEIGHT 

In a mountainous country, the variation of temperature with 
height has to be carefully considered, and its annual course, in 
any case, must be presented. Here also numerical tables are 
preferable to graphs alone. The same is valid for the variation of 
precipitation with height. 

If the number of stations is not too small, and there are con- 
siderable differences in height, a standard temperature-height 
curve and a precipitation-height curve should be constructed. 
From these curves, the anomalies of different places are calculated. 

XVIII. 6. GRAPHICAL REPRESENTATION 

As far as graphical representation is concerned, it is a matter of 
taste which and how many Cartesian diagrams are shown. If these 
appear in addition to numerical tables, they are practical and 
useful. 



ARRANGEMENT OF A CLIMATOGRAPHY 203 

The geographical distribution of at least the most important 
elements (pressure, temperature, precipitation, cloudiness) should 
be represented by maps of isolines, provided the country is not of 
too great ''relief energy" and the mountains do not occupy too 
large a portion (see details XIV. 2). 

For a really mountainous country with a dense network of 
stations up to great altitudes, attention should be focused on the 
construction of maps of isanomals. Only these maps are able to 
reveal the real distribution of temperature and precipitation, and 
particularly the climatic divides, in a mountainous country. The 
method of isanomals eliminates the influence of height on the 
element in question and permits a reasonable interpolation for 
places where no data are available. The anomalies, together with 
the standard curve for the region, yield the actual values of the 
element (see XIV. 2). 

Maps representing the data of the beginning and the end of 
well-defined seasons contribute much to the knowledge of a climate. 
Examples are: season with frost, vegetative period, etc. (See 
VL 2, 3.) 

The addition of an exact definition of the season represented 
should never be overlooked. The frost period, for instance, can 
be defined in different ways: 

1) as the period between the dates when the minimum 
temperature drops below the freezing point the first and 
the last time in the cold season. 

2) as the period between the dates of the first and the 
last day with a mean temperature = 32F. 

3) as the period between the dates at which the average 
curve of the annual course passes the freezing point. 

The vegetative period, also, can be defined variously, as follows: 

1) as the interval of time which lies between the dates 
when the average annual course passes 43 F. 

2) as the space of time between the last and the first "kill- 
ing frost"; this period is called "growing season" in the 
United States. 

The number of these examples could be greatly increased. There- 
fore a clear, exact definition is essential. 

As we have already pointed out, the dates of the beginning 
and the end of a period yield the duration of the period. The 
cartographical representation of "durations" is valuable. Gener- 



204 



METHODS IN CLIMATOLOGY 



ally they are calculated in days. For use in maps they should 
commonly be converted into weeks. These correspond better to 
the usual degree of accuracy of the data and particularly to that 




FIG. 38. Isolines of the dates at which the curve of the annual course 
of temperature rises to 70F in the Mediterranean region. (After V. Conrad) 




FIG. 39. Isolines of the dates at which the curve of the annual course of 
temperature drops below 70F in the Mediterranean region. (After V. Conrad) 

of the trend of the respective isolines. In some cases the unit of a 
month yields the best and the most reliable isolines. 

Figures 38 to 41 illustrate the method of representing the 
dates of beginning and end of defined periods, and the duration 



ARRANGEMENT OF A CLIMATOGRAPHY 



205 



of such periods. (See also XIV. 5. a.) 2 In Figure 38, isolines of 
the dates are seen, on which the curve of the annual course of 
temperature reaches 70F in the Mediterranean region. These 




FIG. 40. Isolines of the duration in weeks of an average daily temperature 
i 70F in the Mediterranean region. (Duration of the hot season). (After 
V. Conrad) 




FIG. 41. Isolines of the number of months with an amount of rain ^ 0.2 
inch. (Isolines of the length of the dry season, Mediterranean region). 
(After V. Conrad) 

isolines are characteristic of the beginning of the hot period; the 
next, Figure 39, shows its end in the same way. On the two maps, 
the average migration of the chosen threshold of temperature can 

2 S. S. Visher, "The Seasons' Arrivals and Lengths," Annals of the Association of 
American Geographers, vol. 33, 1943, pp. 129-134, gives similar maps for the United 
States. 



206 METHODS IN CLIMATOLOGY 

be observed. The average temperature reaches 70F in North 
Africa on May 5, migrates northward and arrives in southern 
Europe, June 10. The retarding effect of the water basin itself 
is well emphasized by means of the closed isoline which corresponds 
to June 5. 3 

Figures 40 and 41 give examples of durations. The first of 
them is derived from Figures 38 and 39, and represents the dura- 
tion in weeks of temperatures equal to or higher than 70F. It is 
the duration of the hot season in this region. On the average, the 
duration increases from north to south. The intensity of this 
increase is, however, greatly diminished by the various effects of 
the surface of the water. 

The discussion of a map of well drawn isolines can be very 
stimulating and can result in a surprising insight into the regional 
climate. 

The last example for this group of problems is offered by Figure 
41, representing the length of the dry season in the Mediterranean 
region. Three features of this map may be stressed: 

1) The "dry season" is well defined by the indication: 
Number of months with an amount of rain ^ 0.2 in. 

2) Parts of the isolines, fhe trends of which are only as- 
sumed, not really based on observations, are drawn as 
broken lines. 

3) The month is taken as a unit, because of the high 
variability especially of small amounts of rain, and the 
great uncertainty in measuring such tiny quantities, 
particularly at times of high temperature and great 
evaporation. 

XVIII. 7. DYNAMICAL CLIMATOLOGY 

This expression was coined by Tor Bergeron, when he extended 
the meteorological methods of the Norwegian school to climatol- 
ogy. 4 Static or statistical climatology represents the average and 
extreme states of atmospheric conditions. Dynamical climatology 
means perhaps development of the average states. No doubt 
many features of each division overlap. Usually, every old- 
fashioned climatography contains, for instance, variabilities which 
belong to the dynamic division. 

1 This date which belongs to this isoline was unfortunately omitted in the figure. 
4 See review by H. C. Willett, "Ground Plan of a Dynamic Meteorology, MWR 
Vol. 59, (1931), 219-223. 



ARRANGEMENT OF A CLIMATOGRAPHY 



207 




208 



METHODS IN CLIMATOLOGY 




ARRANGEMENT OF A CLIMATOGRAPHY 



209 




210 



METHODS IN CLIMATOLOGY 




ARRANGEMENT OF A CLIMATOGRAPHY 211 

A forerunner of these later efforts and their realization is repre- 
sented by the excellent study of Oliver L. Fassig, The Climate of 
Baltimore.* The first part of Fassig's work deals with " static 
climatology," the second with "dynamical climatology." There 
the " types of weather characteristic of the geographical horizon 
of Baltimore" play an important role. A large series of typical 
weather maps on the occasion of special storms, rain areas, frost 
areas, cold waves, blizzards, are published in this highly interesting 
book, which may serve as a pattern for future climatographies. 

Figures 42 to 45 represent an interesting attempt. The average 
distribution of pressure, wind, and temperature of certain months 
(December, March, June, October) which are classified as cold, 
normal, or warm, is shown in the maps. The four months are 
chosen as representative of the four seasons. Although a discus- 
sion of the maps would be out of place here, it is clear from these 
charts that this is the way to understand the development of the 
climatic features of a region. 

From these attempts, it is only one step to the numerical and 
graphical representation of average weather types: e.g., 20 cases 
of Strong foehn wind, east of the Rocky Mountains, are chosen. 
" Strong foehn" has to be defined, for example, by the condition 
that the temperature rises 10F or more within one hour at a certain 
place, representative of the phenomenon. High floods, blizzards, 
hot and cold spells, dry and wet spells can be satisfactorily dis- 
cussed by means of average numerical values and their carto- 
graphical representation. In the writer's opinion, such dynamical 
methods would arrive at the best physical synthesis of the climatic 
elements. In this way, a description of important and character- 
istic climatic phenomena could be given for great regions of the 
continents and for some parts of the oceans, as a guide for long 
range forecasting 6 and for many other purposes. 

6 Oliver L. Fassig, The Climate of Baltimore (Maryland Weather Service, Special 
Pub., vol. 2, Baltimore, 1904). See also Gardener Reed, Jr., in QJRMS 1910, p. 39. 

6 The ideas of a dynamic climatology have also been well expressed in some studies 
by C. F. Brooks and his pupils in " Papers in the Relation of the Atmosphere to Human 
Comfort' 1 (MWR, vol. 53, 1925, pp. 423-437), containing the following articles: C. F. 
Brooks, "The Cooling of Man under Various Weather Conditions"; E, C. Donnelly, 
"Human Comfort as a Basis for Classifying Weather"; F. Howe, "The Summer and 
Winter Weather of Selected Cities in North America"; E. S. Nichols, "A Classification 
of Weather Types"; I. E. Switzer, "Weather Types in the Climates of Mexico, the Canal 
Zone, and Cuba." See also E. S, Nichols, "Frequency of Weather Types at San Jose*, 
Calif.," MWR, vol. 55 (1927), pp. 403, ff. The investigations mentioned attempt to 
avoid the very complicated system of E. E. FedoroflF (MWR vol. 55, 1927, pp. 401 ff.) 
and consider the fact that the principles of the representation should not be more com- 
plex than the phenomenon itself. 



212 METHODS IN CLIMATOLOGY 

XVIII. 8. THE BIOCLIMATOLOGICAL PART 

A bioclimatological section although a short one is expected 
in a climatography. Radiation, cooling power, drying power 
should be mentioned. If no direct observations exist, formulae 
(mentioned above) enable the climatographer to give estimations 
of cooling-and drying-power by means of average values of 
temperature, wind, and humidity. 

If direct observations of radiation are lacking, at least records 
of duration of bright sunshine and descriptions of radiation-climate 
or personal experiences permit a first estimation Of radiation con- 
ditions. 

Some words about acclimatization, endemic diseases, health 
resorts, etc., are usually desired in a more or less complete climatog- 
raphy. The immense hygienic literature presents reliable sources 
from which the climatographer can obtain good information with- 
out assuming too much responsibility. Every climatologist 
should be cautioned against giving his own opinions about subjects 
which are outside the range of his special realm. 7 On the other 
hand, personal experiences with reference to human comfort and 
striking phenomena in the organic and inorganic realm are always 
of the greatest value and should not be suppressed. 

XVIII. 9. DESCRIPTION OF CLIMATIC PHENOMENA 

It should never be forgotten that geography, and even more 
climatology, started with descriptions by early travelers. From 
the books of travels of the famous Venetian Marco Polo to those 
of Alexander von Humboldt, a nearly innumerable series of keen 
travelers with open eyes gave us our first knowledge of the climate 
of remote countries. Nor should modern climatography renounce 
the lively description of climatic impressions drawn from personal 
experience. 

Puka, Albania (42.0N, 19.9E, 2800 ft) is located on the steep 
slopes of the Albanian Alps, which run along the Adriatic Sea coast 
from about NW to SE. Here, in summer, an average of six inches 
of rain falls in eight rainy days. The rain intensity is, on the 
average, about 0.8 in. (For comparison, in Boston the intensity is 

7 An instructive example of this restriction to an objective report (as far as non- 
geophysical influences and effects are concerned) is offered by B. Gutenberg, "Geo- 
physik and Lebewesen," in Lehrbuch d. Geophysik, edited by B. Gutenberg (Berlin, 
1929). 



FIG. 46. Eroded mountain slopes with westerly exposure above Puka, Albania. 

(42 X, 19.9 K, 2800 ft) 



ARRANGEMENT OF A CLIMATOGRAPHY 213 

0.3 in.) This numerical description is interesting and character- 
istic of the mountain climate along the Adriatic coast. A glance 
at Figure 46 gives life to the numbers, and we see the effect of 
these downpours separated from one another by long droughts. 
The slopes with westerly exposure are deeply eroded; desertlike, 
barren ground is before our eyes, and that in a climate where olive, 
fig-tree, and pomegranate thrive and flourish in well protected sites. 

Thus the climatographer should not omit descriptions and suit- 
able pictures which offer the means of making his statistical and 
dynamical features impressive. Books of travels, guidebooks, etc., 
often provide good contributions in this respect, if personal ex- 
perience is lacking. Tree growth and vegetation are in closest 
relation to climate, and architecture also. 

In the Saas Fee valley in the south of Switzerland (one of the 
most beautiful valleys of the Alps), the hay barns stand on high 
stone pillars. Even if observations of precipitation did not exist, 
from the height of the pillars we should know not only that copious 
snowfalls occur in this region but also the average maximum depth 
of snow on the ground. 

In the whole Mediterranean region from Italy to North Africa, 
narrow lanes are usual so narrow that two people can hardly pass 
one another. High altitudes of the sun, combined with frequently 
cloudless skies and a dry, clean atmosphere with low turbidity 
factors characterize the weather of the summer half-year. The 
architectural effect resulting from these scientifically described 
radiation conditions is narrow lanes. Conversely, we conclude 
that high radiation is present when narrow lanes are the charac- 
teristic architecture. Arcades along the streets signify frequent 
downpours as well as intense radiation. High gable roofs are often 
the architectural expression of habitually intense snowfalls. 

High walls encircling precious orchards and vineyards are 
characteristic of the northern coast of the Mediterranean, protect- 
ing the crops against the mistral and the bora and similar bad and 
frequent storms. In Normandy and Brittany the traveler is 
surprised to see miles and miles of high hedges, which surround 
fields and farms. One concludes, here again, that frequent storm- 
like winds are found where such walls and hedges are typical. 8 

Numerous settlements high on the slopes and a deserted bottom 
of the valley indicate frequent floods, and also intense inversions 

8 The reader will find excellent examples of climatically characteristic photos in the 
Atlas of American Agriculture, in the sections on "Soil" and "Natural Vegetation." 



214 METHODS IN CLIMATOLOGY 

of temperature in fall and spring. Manners and customs of man- 
kind are often highly influenced by climatic features. It was 
natural that the traveled R. DeCourcy Ward was enthusiastic 
about the necessity of colorful descriptions; he went so far as to 
speak of a " car- window climatology." None other than J. Hann 
always enjoyed getting lively descriptions to cover his numerical 
skeletons. The many descriptions in his famous regional climatog- 
raphy of the world attest these efforts. 



CONCLUSION 

THE READER who runs over the pages of this book is perhaps 
disappointed in not finding this or that special method. 
In the first pages, it was shown that the number of de- 
rived elements is not limited theoretically; thus completeness is 
unattainable. 1 

Moreover, it is intended to present only a system of methods 
which facilitates the step from the qualitative to the quantitative. 
Many ways are offered for attaining this purpose, and they all 
should lead not only to an exact and comparable description of the 
climate but, in the end, to a physical explanation. 

1 The reader will find a rich source of different methods of representation in the 
monumental work: Atlas of American Agriculture: Physical Basis including Land Relief, 
Climate, Soils, and Natural Vegetation of the United States (Washington, 1936), prepared 
under the supervision of O. E. Baker, with contributions from the Weather Bureau, 
Willis R. Gregg, H. G. Knight, Frederick D. Richey, F. A. Silcox, A. G. Black. 



215 



APPENDIX I 



FORMS OF INDIVIDUAL CLIMATIC TABLES 

THE FOLLOWING form is that of the " Climatic Summary of the 
United States" published by the U. S. Weather Bureau. The 
form is slightly modified with respect to its heading. Two columns 
have been added: " Depth of Snow on the Ground " and " Estimated 
Average Wind Velocity." The first columns can be filled with informa- 
tion, based on the regular observations of cooperative stations in the 
United States. 

The average wind velocity is not recorded by the cooperative ob- 
servers. An estimate of the average of the whole day could easily be 
made according to the following simplified scale: 

= calm to light 

1 = gentle 

2 = strong to stormy 

Too little attention is paid to the wind in climatological publications. 

BLANK FORM OF A CLIMATIC TABLE 
(Taken from the "Climatic Summary of the United States," U. S. Weather Bureau) 

Name of the Place Country 

< == X = H = feet (meters) 

Period: 

[Remarks about special local conditions, period, instruments, etc.] 





Temperature 


Precipitation 


Relative 
Humidity 


Sun- 
shine 


Wind 


















8 




^ 


Snow 


































E 


s 


jJJ 
















































-M 









































.S 


CO 


S 




to 

3 


















o 












W 






-H 


*<H 


flj 





















c 


**^ 




rt 




rt 




S 


bo 
ca 




O 




Q 


^ 




J5 


js 














O 

t? 


c 


s 
1 


<D 

3 

*O 
0) 

f 


1 Absolute Maximum 


1 Average of the Minim 


Absolute Minimum 


Si 

jE: 



CO 
4> 

,bf) 


Lowest Monthly Aver 


1 
1 


1 Number of Days with 


0) 

rS 
1 

1 


(U 

s 

c 



A 


Average Amount 


Greatest Amount in 2 


1 

O 

o 

S 
ex 

Q 


E 

r 


^ 
f 

c 


i 


0> 

i 


1 Average Hours 


Percentage of Possible 


Prevailing Wind Direc 


Estimated Average W 



























* According to daily estimations: calm to light, 1 ** gentle, 2 = strong to 
stormy. 

216 



APPENDIX I 217 

Julius Harm suggested the following content of a climatological table. 
The form is in principle identical with that of the U. S. WeatherBureau. 
A great number of climatologists, perhaps the majority, have made use 
of the following suggestions as far as available data permitted. 

A. Temperature 

a. Averages for the three observation times 

b. True monthly means 

c. Extreme monthly and annual means for the period in question 

d. Averages of the interdiurnal variability 

e. Mean daily periodical range (if records of a thermograph are 
available) 

f. Mean daily extremes 

g. Mean absolute extremes and their difference 
h. Absolute extremes 

i. Average frequencies of certain differences between consecu- 
tive daily means (e.g. differences: 3F to 6; 7 to 10F etc.) 

B. Vapor pressure 

C. Relative humidity (morning, early afternoon, daily mean) 

D. Cloudiness (morning, early afternoon, daily mean) 

E. Sunshine duration (hours, percentage) 

F. Precipitation 

a. Average amounts 

b. Relative variability 

c. Snow (and snow mixed with rain), amounts 

d. Depth of snow on ground 

G. Wind 

a. Resultant direction 

b. Resultant velocity (vectorial) 

c. Average velocity, disregarding the direction 

d. Steadiness 
H. Air pressure 

a. Averages 

b. Average monthly and annual range 
I. Number of days 

a. Clear. Overcast 

b. Without sunshine 

c. With precipitation S 

d. With hail 

e. With snowfall 

f. With thunderstorm 

g. With storm ( > Beaufort 6) 

h. With frost (minimum below freezing point) 
i. With fog 

This table should be completed at least by a frequency distribution 
(per cent) of 8 wind directions and calm for the months and the year. 
Other frequency distributions could be added according to the purpose 
of the climatography. 



APPENDIX II 



(See XII.2.b and the "Intermediate Chapter") 
1 234567 








1.000 


.707 


.577 


.500 


.447 


.408 


.378 


.354 


.333 


10 


.316 


.302 


.289 


.277 


.267 


.258 


.250 


.243 


.236 


.229 


20 


.224 


.218 


.213 


.209 


.204 


.200 


.196 


.192 


.189 


.186 


30 


.183 


.180 


.177 


.174 


.171 


.169 


.167 


.164 


.162 


.160 


40 


.158 


.156 


.154 


.152 


.151 


.149 


.147 


.146 


.144 


.143 


50 


.141 


.140 


.139 


.137 


.136 


.135 


.134 


.132 


.131 


.130 


60 


.129 


.128 


.127 


.126 


.125 


.124 


.123 


.122 


.121 


.120 


70 


.120 


.119 


.118 


.117 


.116 


.115 


.115 


.114 


.113 


.113 


80 


.112 


.111 


.110 


.110 


.109 


.108 


.108 


.107 


.107 


.106 


90 


.105 


.105 


.104 


.104 


.103 


.103 


.102 


.102 


.101 


.101 


100 


.100 


.100 


.099 


.099 


.098 


.098 


.097 


.097 


.096 


.096 


110 


.095 


.095 


.094 


.094 


.094 


.093 


.093 


.092 


.092 


.092 



218 



APPENDIX III 

AUXILIARY TABLE, CALCULATING THE EQUIVALENT TEMPERATURE FROM 
AIR-TEMPERATURE (/C) AND RELATIVE HUMIDITY (AFTER F. LINKE) 

(See IX.2) 









K(t t b) 






Air-Pressure in Millimeters 


of Mercury 




*C 


780 


760 


740 


720 


700 


-45 


0.11 


0.11 


0.12 


0.12 


0.12 


-40 


0.20 


0.20 


0.21 


0.22 


0.23 


-35 


0.33 


0.34 


0.34 


0.35 


0.36 


-30 


0.57 


0.59 


0.61 


0.62 


0.64 


-25 


0.95 


0.97 


1.00 


1.02 


1.05 


-20 


1.55 


1.59 


1.64 


1.68 


1.72 


-18 


1.87 


1.92 


1.98 


2.04 


2.09 


-16 


2.29 


2.34 


2.40 


2.47 


2.52 


-14 


2.75 


2.81 


2.88 


2.97 


3.02 


-12 


3.26 


3.37 


3.45 


3.54 


3.61 


-10 


3.90 


4.01 


4.11 


4.23 


4.33 


- 8 


4.72 


4.87 


5.00 


5.12 


5.26 


- 6 


5.80 


5.87 


6.00 


6.20 


6.40 


- 4 


6.77 


6.90 


7.13 


7.33 


7.60 


- 2 


7.90 


8.00 


8.25 


8.50 


8.75 





9.09 


9.32 


9.58 


9.84 


10.1 


2 


10.5 


10.7 


10.9 


11.2 


11.5 


4 


12.0 


12.2 


12.5 


12.9 


13.3 


6 


13.9 


14.2 


14.6 


15.1 


15.5 


8 


15.8 


16.2 


16.6 


17.2 


17.6 


10 


18.1 


18.6 


19.1 


19.6 


20.1 


12 


20.6 


21.1 


21.7 


22.3 


22.9 


14 


23.4 


24.2 


24.8 


25.5 


26.2 


16 


26.6 


27.3 


28.2 


28.9 


29.7 


18 


30.2 


30.8 


31.7 


32.7 


33.6 


20 


34.2 


35.1 


36.1 


37.0 


38.2 


21 


36.4 


37.2 


38.4 


39.4 


40.4 


22 


38.6 


39.5 


40.6 


41.7 


43.0 


23 


40.8 


41.9 


43.4 


44.5 


45.6 


24 


43.5 


44.5 


46.0 


47.3 


49.4 


25 


46.7 


47.5 


48.8 


50.2 


51.6 


26 


49.0 


50.1 


51.6 


53.0 


54.6 


27 


51.9 


53.2 


54.6 


56.9 


57.8 


28 


54.9 


56.3 


57.9 


59.6 


61.2 


29 


58.3 


59.7 


61.4 


63.0 


64.7 


30 


61.9 


63.5 


65.2 


67.0 


68.8 


31 


65.4 


67.2 


69.1 


71.1 


72.6 


32 


69.2 


71.2 


72.9 


75.2 


76.7 


33 


73.3 


75.2 


77.1 


79.3 


81.3 


34 


77.6 


79.6 


80.4 


84.0 


85.0 


35 


82 


84 


86 


89 


91 


36 


87 


89 


91 


94 


96 


37 


92 


94 


96 


99 


102 


38 


97 


100 


102 


105 


108 


39 


103 


105 


108 


111 


114 


40 


108 


110 


113 


116 


120 



219 



APPENDIX IV 

THE DAYS OF AN ORDINARY YEAR, NUMBERED CONSECUTIVELY, 
BEGINNING JANUARY FIRST 





Jan. 


Feb. 


Mar. 


Apr. 


May 


June 


July 


Aug. 


Sept. 


Oct. 


Nov. 


Dec. 


1 


1 


32 


60 


91 


121 


152 


182 


213 


244 


274 


305 


335 


2 


2 


33 


61 


92 


122 


153 


183 


214 


245 


275 


306 


336 


3 


3 


34 


62 


93 


123 


154 


184 


215 


246 


'276 


307 


337 


4 


4 


35 


63 


94 


124 


155 


185 


216 


247 


277 


308 


338 


5 


5 


36 


64 


95 


125 


156 


186 


217 


248 


278 


309 


339 


6 


6 


37 


65 


96 


126 


157 


187 


218 


249 


279 


310 


340 


7 


7 


38 


66 


97 


127 


158 


188 


219 


250 


280 


311 


341 


8 


8 


39 


67 


98 


128 


159 


189 


220 


251 


281 


312 


342 


9 


9 


40 


68 


99 


129 


160 


190 


221 


252 


282 


313 


343 


10 


10 


41 


69 


100 


130 


161 


191 


222 


253 


283 


314 


344 


11 


11 


42 


70 


101 


131 


162 


192 


223 


254 


284 


315 


345 


12 


12 


43 


71 


102 


132 


163 


193 


224 


255 


285 


316 


346 


13 


13 


44 


72 


103 


133 


164 


194 


225 


256 


286 


317 


347 


14 


14 


45 


73 


104 


134 


165 


195 


226 


257 


287 


318 


348 


15 


15 


46 


74 


105 


135 


166 


196 


227 


258 


288 


319 


349 


16 


16 


47 


75 


106 


136 


167* , 


197 


228 


259 


289 


320 


350 


17 


17 


48 


76 


107 


137 


168 


198 


229 


260 


290 


321 


351 


18 


18 


49 


77 


108 


138 


169 


199 


230 


261 


291 


322 


352 


19 


19 


50 


78 


109 


139 


170 


200 


231 


262 


292 


323 


353 


20 


20 


51 


79 


110 


140 


171 


201 


232 


263 


293 


324 


354 


21 


21 


52 


80 


111 


141 


172 


202 


233 


264 


294 


325 


355 


22 


22 


53 


81 


112 


142 


173 


203 


234 


265 


295 


326 


356 


23 


23 


54 


82 


113 


143 


174 


204 


235 


266 


296 


327 


357 


24 


24 


55 


83 


114 


144 


175 


205 


236 


267 


297 


328 


358 


25 


25 


56 


84 


115 


145 


176 


206 


237 


268 


298 


329 


359 


26 


26 


57 


85 


116 


146 


177 


207 


238 


269 


299 


330 


360 


27 


27 


58 


86 


117 


147 


178 


208 


239 


270 


300 


331 


361 


28 


28 


59 


87 


118 


148 


179 


209 


240 


271 


301 


332 


362 


29 


29 




88 


119 


149 


180 


210 


241 


272 


302 


333 


363 


30 


30 




89 


120 


150 


181 


211 


242 


273 


303 


334 


364 



31 



31 



90 



151 



212 243 



304 



365 



220 



INDEX 



Abbe, E., 134, 135, 136, 137, 138, 140 

"Above normal," 37 

Absolute degrees of temperature, 200 

Absolute drought, 123 

Absolute extremes, 17, 30 

Absolute frequency, 19 

Absolute maximum, 19, 28 

Absolute minimum, 19, 28 

Absolute probability of precipitation, 53 

Absolute temperature, 200 

Absolute variability, 31 

Absolutely homogeneous climatological 
series, 130 ff. 

Acclimatization, 212 

Actual isohyets, 183 

Actual temperatures, 172 

Air mass, continental, 193 
index of continentality, 193 
maritime, 193 

Air mass analysis, vs. windroses, 188 

Air mass climatology, ix, 193 

Air temperature, "true," 13 

Amplitude, relative, 78 

Angot, A., 96, 121 

Angstrom, A., 115 

Annual course of the amounts of precipi- 
tation, 120 

Anomalies, methods of, ix, 170, 173 ff. 
of precipitation, 180 

Antipleions, 178 

Aperiodic variations, 202 

Aperiodical range of temperature, 88, 89 

Arctowski, H., 178 

Aridity, index of, 127 

Arithmetical mean, 30 

Arkin, H., 22, 45, 46, 87 

Array, 24 

Artificial heating, 94, 95, 97 

Ash, D. H., 113 

Aspiration-thermometer, 13, 14 

Association, degree of, 151 

Asymmetry of the annual course of tem- 
perature, 99 

Asymmetry of frequency curves, 44 ff. 

Atmospheric pressure, 100 

Average absolute extremes, 30 

Average variability, 31, 46 

Azimuth of wind direction, 103 



Bacon, F., viii 

Baker, O. E., 215 

Bavendick, F. J., 32 

Beaufort, Sir Francis, 186 

Beaufort numbers, equivalent of, 105 

Bell-shaped curve, 34 

Bergeron, T., 193, 206 

Berlage, H. P., Jr., 102 

Best, A. C., 14 

Biel, E., 51 

Bigelow, F., 84, 89, 90, 121 

Bilaterally limited variates, 44 

Binomial coefficient, 69 

Bioclimatological part of a climatography, 

212 

Birkeland, B. J., 147 
Bjerknes, V., 169 
Black, A. G., 215 
Blair, T. A., 97 
Blue Hill Observatory, x, 13, 88, 116, 

117 

Book of normals, 51 
Bottom layers, 15 
Braak, C., 182 
Break of series, 12 
Brightness of the sky, 57 
British Meteorological Office, 7, 8 
Brooks, C. E. P., 87, 88, 89, 160, 161 
Brooks, C. F., 13, 15, 46, 84, 95, 113, 114, 

122, 185, 199, 200, 211 
Bruckner, E., 147 
Bruckner-cycle, 147 
Brunt, D., 21, 78, 114 
Brunt's continentality factor, 196 
Brunt's criterion, 78 
Buttner, K., 113 

Calms, 105 

"Car window climatology," 214 

Central heights of the class intervals, 19 

Central Institute (Office), 7 

Change of signs, 134 

Chapman, F. H., 37 

Characteristics, elementary, 17, 24 

system of, 17 

higher, 17 

primitive, 17 



221 



222 



METHODS IN CLIMATOLOGY 



Class intervals, 19 

Clayton, H. H., 8, 132 

Clear days, 115 

Climate, definition, 1, 2 

Climatic divide, ix, 149, 183, 203 

Climatic elements, synthesis of, 211 

Climatic exposure, 176 

Climatic limits of timber forest, steppe, 

desert, 196 

Climatic phenomena, description of, 212 
Climatic regions, coherent, ix, 148 

incoherent, 148 
Climatic tables, 216 
Climatography, 1, 199 

arrangement of, 199 

bioclimatological part, 212 
Climatological averages, reduction to a 

certain period of, 139 
Climatological elements, combined, 4 

derived, 4 

groups of, 3 

primitive, 4 

Climatological normals, 148 
Climatological series, absolutely homo- 
geneous, 131 ff. 

comparableness of, 3 

definition of the relative homogeneity 
of, 134 

discontinuities of, 12 
Climatology, 1 

air-mass, ix, 193 

dynamical, 206 ff. 

static, 200 
Climogram, 107 
Cloudiness, 115 

a bilaterally limited element, 44 

and relative sunshine duration, 55, 115 

reduction of, to the "true mean," 91 
Cloudy days, definition, 115 
Codex, International Meteorological 83 
Coefficient, binomial, 69 

of correlation, 151 

of variation, 38 

Coherent climatic regions, ix, 148 
Cold days, spells of, 95 
Colton, R. R., 22, 45, 46, 87 
Combined elements, 4, 107 
Committee, International Meteorological, 

6 
Comparableness of Climatological series, 

2,3 

Comparison, graphic, 166 
" Computer's Handbook/' 160 
Conferences, International Meteorological, 
6 



Conrad, V., 2, 5, 13, 37, 38, 49, 50, 51, 52, 
57, 59, 61, 62, 64, 65, 75, 91, 92, 97, 
99, 100, 104, 113, 124, 126, 134, 135, 
136, 138, 144, 146, 148, 149, 174, 176, 
178, 180, 181, 190, 191, 204, 205 
Constituents of a wave, 77 
Continental air masses, 193 
Continentality, ix, 4, 195 

index of, based on air mass frequency, 

193 

Continentality factor, Brunt's, 196 
Convergence of series, 78 
Cooling power, 112 ff., 212 

new formulas for, 1 13 

and physiological processes, 114 
Coolings, 49 
Cooperative observers and stations, 9,11, 

15, 17, 87, 115, 125 
Cornu's theorem, 39 

Correction to the "true mean," 88, 89, 90 
Correlation, ix, 150 ff. 

example of calculating, 153 

linear, ix, 150 

partial, 161 

simplifications in computing, 155 
Correlation coefficient, 151, 164 

partial, 164 
Crestani, G., 92 
Cumulated temperatures, 96 
Curve fitting, 58 
Curve parallels, 160 
Czuber, E., 19 

Daily course of temperature, 81 

Daily mean, 83 

Decay curves, 62 

Deciles, 24 

Degree of association, 151 

Degree days, 97 

Degrees absolute, 200 

Deluc, J. A., 1 

Denison, F. R., 15 

Departure, 31 

Derived elements, 4 

Description of climatic phenomena, 212 ff. 

Desert and steppe, climatic borderline 

between, 196 ff. 
Deviation, 31 
Differences, method of, 139 

quasi-constancy of, 131 
Dines, W. M. ( 160 
Dinies, E., 193 
Direction of wind, 103 
Directional quantities, 48 
Discontinuities of Climatological series, 12 
Dispersion of a variate, 38 



INDEX 



223 



Distribution, normal, 31 

of temperature in mountainous regions, 

170 

Divides, climatic, 149, 182, 203 
Donnelly, E. C, 113, 114, 211 
44 Dot charts," 153, 160 
44 Dot diagram,' 1 157 
Drought, absolute, 123 

partial, 124 
Dry season, 205 
Dry spells, 123, 124 
Drying power, 111 
Duration, 94 

cart pluviometrique relatif, 120, 121 
Effectiveness of precipitation, 196, 198 

index of, 198 
Ekhart, E., 126 
Ekholm, N., 87 

Elementary characteristics, 17, 24 
Elements, combined, 4 

derived, 4 

primitive, 4 
Endemic diseases, 212 
Ephemerides Societatis Meteorologicae 

Palatinae, 5 
Equivalent temperature, 4, 108 ff. 

auxiliary table calculating the, 219 
Equivalents of Beaufort numbers, 105 
Eredia, F., 107 
"Etesians," 190 

Evaluation of Fourier's series, 76 
Excess, 38, 41 

positive, 38 
Exner, F. M., 162 
Exponential curve, 63 
Exposure, climatic, 176 

of the thermometer shelter, 15 
41 Extremely above normal," 37 
41 Extremely subnormal," 37 
Extremes, absolute, 17 

average absolute, 30 

mean daily, 30 

of temperature, 87 

Fassig, O. L., 202, 207, 208, 209, 210, 211 
Favorable events, 35 
FedorofT, E. E., 211 
Fergusson, S. P., 118 
Fisher, L. C, 32 
Fisher, R. A., 21, 29,41,46 
Fog, 116 

Forbes,]. D., 178 
11 Force majeure," 36 
Forecast, 149 
Fourier's series, 70, 78, 79 
evaluation of, 76 



Frequencies of wind directions, 103 
Frequency, absolute, 19 

relative, 19 
Frequency curve, 21, 23 

asymmetry of the, 44 ff. 
Frequency distribution, 19, 201 
Frogner, E., 147 
Frost days, 95 
Frost period, 4, 93, 94, 95 

different definitions, 203 

Galilei, Galileo, 1 

Gal ton, F., 160 

Gauss, K. F., 34 

Gaussian distribution, 33 

Gaussian law, 147 

Geiger, R., 14, 199 

General equation of a conic section, 64 

Geographical latitude and temperature, 

177 

Germination, temperature of, 94 
Gherzi, E., 53, 127 
Gotz, F. W. P., 4, 107 
Goodnough, X. H., 132 
Gorczynski, W., 127, 195, 196 
Graphic comparison, 166 
Graphic methods of representation, ix, 202 
"Greatly above normal," 37 
41 Greatly subnormal," 37 
Gregg, W. R., 215 
Grouping the data, 19 
Growing season, 95, 203 
Gutenberg, B., 212 

Hand, I. F., 167 

Hann, J. von, viii, 80, 124, 145, 149, 171, 

177, 178, 179, 214, 217 
Harmonic analysis, 70 ff. 

Pollak's tables to, 73 
Haurwitz, B., 193 
Health resorts, 212 
Heat balance of the human body, 114 
Hellmann, G., 7, 52 

Helmert, F. R., criterion, 134, 135, 138, 140 
Higher characteristics, 17 
Hildebrandsson, H. H., 7 
Hill, L., 112, 113 
Histogram, definition, 21 

of rain, 127 
Homogeneity, 200 

relative, of climatological series, defi- 
nition, 134 
Homogeneous climatological series, 130, 

131 ff. 
Hot days, spells of, 95 



224 



METHODS IN CLIMATOLOGY 



Hot season, length of, 205 

Howe, F., 211 

Human body, heat balance of the, 1 14 

Human skin, temperature of, 1 14 

Humboldt, A. von, 212 

Huntington, E., 153 

Hyperbola, different forms, 64 

Ice days, 95 

spells of, 96 

I MC = International Meteorological Com- 
mittee, 84 

Incoherent climatic regions, 148 
Index, of aridity, 127 

of continentality, based on air masses 
frequency, 193 

of precipitation-effectiveness, 198 
Inhomogeneity of climatological series, 130 
Integration, numerical, 82 
Intensity of rain, 54, 122 
Interannual variability, 50 
Interdiurnal variability, 48 
Interhourly variability, 50 
International Climatological Register, 10, 

81 
International Conference of Maritime 

Meteorology, 6 

International Meteorological Codex, 7, 83 t 
International Meteorological Committee, 

6, 83, 84, 105 
International Meteorological Conference 

(Utrecht, 1923), 8 
International Meteorological Congress 

(Vienna, 1873), 6 

International Meteorological Organiza- 
tion, 14 

(Utrecht, 1874), 9 

Interpolation of missing observations, 148 
Interseasonal variability, 50 
Intersequential variability, 50 
Introduction to a climatography, 199 
Inversion of temperature, 13, 65, 170 
Isanomals, ix, 126, 173 ff. 

maps of, 203 

method of, 174 
Isogram, 166 ff. 
Isohels, 167 
Isohyets, 167, 180 

actual, 183 
Isolines, maps of, 166 ff. 

projections for maps of, 169 

rules for drawing, 167 
Isonephs, 167 
Isopleths, 166 
Iso-stereograms, 167 
Isotherms, 167 



Johansson, O. V., 196 

Kelvin, Lord, 200 

Killing frost, 31, 95, 203 

Kincer, J. B., 9 

Knight, H. G., 215 

Knoche, W., Ill 

Koppen, W., 46, 53, 54, 99, 115, 196, 197, 

199 

Kubitschek, O., 64, 126 
Kuhlbrodt, E., 189 
Kurtosis, 41 

Lambert's formula, 101, 102, 104 

Landsberg, H., 193 

Lapse rate, uniform average of, 170, 171, 

172, 182 
Latitude, geographical, and temperature, 

177 

Least squares, method of, 60, 65, 71, 87 
Left skewness, 44 
Lehmann, H., 113 
Linear correlation, ix, 150 ff. 
Linke, F., 108, 109, 219 
London, Meteorological Office, 51 
Long-range forecast, 2, 149 
Lower quartiles, 28 

Lutschg, O., 182, 183, 184 

* 

McAdie, A., 84, 117 

Macelwane, J. B., 201 

Mahalanobis, P. C., 38 

Manila, Observatorio Central de, 7 

"Mannheimer Akademie," 5 

Map of isolines, 166 ff. 

Mapping temperature conditions of a 

mountainous region, 170, 171 
Maritime air masses, 193 
Martonne, E. de, 127 
Maurer, J., 171, 174 
Maximum, absolute, 19, 28 
Mean, weighted, 85 
Mean daily extremes, 30 
Mean departure, 31 
Mean deviation, 31 
Median, 14 

Meinardus, W., 166, 171, 177, 178 
Meions, 178 

MSL = mean sea level, 8 
Meteorological Codex, International, 83 
Meteorological Committee at Vienna 

(1927), 105 
Meteorological Glossary (London), 1, 31, 

105, 123, 148, 152, 186 
Meteorological Office (London), 51, 148, 

152, 156 



INDEX 



225 



Meteorological Register, 9, 17 
Methods, of anomalies, 170, 174 

of climatological observations, 12 

of differences, 139 

of least squares, 60 

of quotients, 142 

semi-average, 61 
Meyer, H., 79, 115 
Microclimatology, 14 
Midpoint of the class interval, 19 
Miller, G., 193 
Miller, J. K., 169 
Minimum, absolute, 19, 28 
Missing observations, interpolation of, 148 
Modal class, definition of, 21, 22 
Mode, 21 
Morikofer, W., 4 
Mohn, H., 166 
Monographs, 201 

Monotony of high temperatures, 21 
Months of equal length, reduction to, 120 
Mountainous regions, mapping tempera- 
ture conditions of, 170, 171 

change of precipitation with height in, 

202 

Mount Washington Observatory, 18, 22, 
30,49 

Namias, J., 193, 201 
Network of stations, 5, 6 
Neuberger, H., 62, 118 
Nichols, E. S., 211 
Nipher, F. E., 15 
Non-coherent climatic regions, ix 
44 Normal, 11 37 
Normal distribution, 31 
Normal period, length of a, 141, 146 
Normal station, 141 
Normal values, 37, 51 
Normals, book of, 51 
climatological, 148 
Norwegian school, 206 
Noyes, G. H., 117, 119 
Numerical integration, 82 
Numerical series, smoothing of, 68 

Observatorio Central de Manila, 7 
Oceanity, 195 

Organization, International Meteorolog- 
ical, 14 

Orographic microclimatology, 14 
Overlapping sums, 69 



Partial drought, 124 
Peakedness, 41 
Pearson, K., 45, 152 
Percentiles, 39 

Period, auxiliary table for calculating the 
length of a, 200 

normal, 141 if. 
Periodic phenomena, 170 
Periodic variations, 202 
Periods of snow cover, 126 
Phase angle, 70 
Physiological processes and cooling power, 

114 

Pleions, 178 
Point cloud, 157 
Pollak, L. W., viii, 17, 19, 44, 72, 74, 80 

102, 133, 157, 165, 167 
Pollak's tables to harmonic analysis, 73 
Polo, Marco, 212 
Positive excess, 38 
Potential water power, 4 
Precipitation, 118 

absolute probability of, 53 

amounts of, within certain periods, 119 

annual course of the amounts of, 120 

anomalies of, 180 

on an average day of the month, 121 

change with height, 202 

duration of, 54 

effectiveness of, 196 ff. 

method of measuring, 4, 15 

profiles of, 183 

variability of, vs. amounts of, 185 
Pressure, atmospheric, 100 

of water vapor, 100 
Primitive characteristics, 17 
Primitive elements, 6 
Probability, 35 

absolute, of precipitation, 53 

of a day with precipitation, 118 
Probability integral, 35 
Probable error, 36 

ratio between and correlation coeffi- 
cient, 152 

calculation of (auxiliary table), 218 
Projections for isoline maps, 169 

Quartiles, 24 
Quasi-constancy, 150 

of differences, 131 ff. 

of ratios, 132 ff . 
Quotients, method of, 142 

explanation of the method, 143 



Partial correlation, ix, 161, 164 
Partial correlation coefficients, 164 



Radiation, 212 
local influences of, 15 



226 



METHODS IN CLIMATOLOGY 



Rain, histogram, 127 

intensity, 54, 122 

spells, 124 

Rain measurements in Palestine, 1 
Rainfall, 122 
Rain-gauge, diameter of, 16 

heights of the rim of, 16 

in India, 400 B.C., 1 

shielded, 15 
Random samples, 53 
Range of the extremes, #-years average of, 

30 

Range of temperature, aperiodical, 88, 89 
Range of variation, 17, 19, 25 
Rareness of a climatic event, 36 
"Ratio of variation," 52 
Ratios, quasi-constancy of, 132 
Reduction, to a given level (temperatures), 
171 

of climatological averages to a certain 
period, 139 

of cloudiness to the "true mean," 91 

to the "true mean," 81 

to a normal period, limits of the method 
of, 144 

of wind velocity to the "true mean," 92 
Reed, G., Jr., 211 
Reed, W. G., 95 
Register, International Climatological, 10, 

81 

Regression coefficient, 160 
Regression equation, ix, 157 ff. 
Relative amplitudes, 78 
Relative frequency, 19 
Relative humidity, 4 

Relative pluviometric coefficient, 120, 121 
Relative shunshine duration and cloudi- 
ness, 115 

Relative temperatures, 98 
Relative variability, 51, 122 
"Relief energy," 171, 173, 176 
Renk, J. J., 201 
"Roseau mondial," 2, 5, 7 
Resultant run of the wind, 101, 104 
Resultant wind velocity, 101, 105 
Richey, F. D., 215 
Riesbol, H. S., 15 
Rietz, H. L., 46, 64 
Right skewness, 44 
Rotch, A. L. f 15 
Run of the wind, 104 

Sable, E., 97 

Sample of the variate, 18 
Scalar quantities, 81 
Scatter of a variate, 38 



Schell, I. I., 149 
Schreier, O., 138 
Schuster, Sir Arthur, 80 
Season, auxiliary table for calculating the 
length of a, 220 

cold, normal, warm, 207-210 

growing, 95, 203 

length of dry, 205 

length of hot, 205 

of snow-cover, 125 
"Secular Station," 13, 141 
Self-registering instruments, records of, 

202 

Semi-average method, 61 
Sequence of signs, 134 
Series, absolutely homogeneous climato- 
logical, 131 ff. 

break of, 12 

convergence of, 78 

discontinuity of, 12 
Sextiles, 39 
Shaw, Sir Napier, 5, 7, 31, 100, 160, 161, 

169, 186, 187 
Shelter, thermometer, 15 
Shielded rain-gauge, 15 
Showalter, A. K., 193 
Signs, change of, 134 

sequence of, 134 
% Silcox, F. A., 215 
Skewness, 42, 44 
Sky, brightness of the, 57 
Smithsonian tables, 101, 105 
Smoothing of numerical series, 58, 68 ff. 
Snow, 124 

depth, of freshly fallen, 125 
Snow cover, 95 

and altitude, 180 

duration of a, 125 

periods of, 126 

season of, 125 
Snowfall, days with, 119 

duration value of, 126 
Special statistical characteristics, 30 
Spells, of cold days, 95 

of dry and wet days, 124 

of hot days, 95 

of ice days, 96 

of tropical days, 96 

of summer days, 96 
Spilhaus, A. F., 168 
Spitaler, R., 196 
Standard curves, 175 
Standard deviation, 31, 124 

of the arithmetical mean, 33 
Standard distribution of an element, 67, 
174 



INDEX 



227 



Standard equation, 157 

Standard period for climatological nor- 
mals, 148 

Static climatology, 200 

Stations of different order, 5 ff. 

Statistical characteristics, 17 

Steadiness of the wind, 106, 189 

Steinhauser, F., 25, 126, 201 

Steppe, timber forest, and desert, climatic 
borderlines between, 196 ff. 

Stevenson's screen, 13 

Stone, R. G., 113, 126, 193 

Stratification of the air, 14, 15 

Streamlines, maps of, 188 

" Subnormal ," 37 

SOring, R., 177, 178, 179 

Summer days, spells of, 96 

Sunshine, duration of, vs. altitude, 180 
and cloudiness, 55 

Supan, A., 196, 197 

"Superior force," 36 

Superposition of waves, 71 

Switzer, I. E., 211 

Swoboda, G., 196 

Synthesis of climatic elements, 211 

Tables to the harmonic analysis, 73 
Temperature, absolute, 200 

actual, 172 

aperiodical range of, 88, 89 

of artificial heating, 94, 95 

asymmetry of the annual course of, 99 

change with height, 202 

cumulative, 96 

daily course of, 81 

distribution of, in mountainous regions, 
170 

duration of a certain, 94 

equivalent, 108 

and geographical latitude, 177 

of germination, 94 

of the human skin, 1 14 

inversion, 13, 65, 170 

mapping of, in a mountainous region, 171 

reduction of, to a given level, 171, 173 

relative, 98 

surges, 49, 50 

threshold of a certain, 92 

true, of the air, 13 
Temperature-height curve, 174 
Tercentesimal, 200 
Thermal stratification, 14, 15 
Thermomeions, 180 
Thermometer, invention of, 1 
Thermometer shelter, 15 



Thermopleions, 178 
Thornthwaite, C. W., 198 
Thresholds of temperature, 92 
Timber forest and steppe, climatic border- 
line between, 196 ff. 
Times of the extremes, 78 
Tippett, L. H. C., 34 
Trajectories, 188 
Tree-growth analysis, viii 
Trewartha, G. I., 186, 193 
Triangulation, climatological, 138 
Tropical days, spells of, 96 
"True azimuth" of wind direction, 103 
"True Mean," correction of the, 81 ff. 
reduction of wind velocity to the, 92 
"True phase angle," 75 
True temperature of the air, 13 
Turning point of a curve, 66 

Unilaterally limited variates, 39 

United States Weather Bureau, viii, 7, 9, 

11, 15, 16, 17, 84, 89, 97, 115, 121, 

125, 201, 216, 217 
Upper quartile, 28 

Variability, absolute, 51, 122 

average, 31, 46 

interannual, 50 

interdiurnal, 48 

interhourly, 50 

interseasonal, 50 

intersequential, 50 

relative, 50, 122 
Variability of precipitation vs. amounts of 

precipitation, 165 
Variance, 38, 78 
Variate, 17 

bilaterally limited, 39 

dispersion of a, 38 

unilaterally limited, 39 
Variations, aperiodic, 202 

periodic, 202 

range of, 19 

ratio of, 52 
Varney, B. M., 50 
Vectorial quantities, 81 
Vegetative period, 4, 95, 174, 176, 203 
Visher, S. S., 205 

Wagner, A., 57, 201 

Wagner, H., 171 

Walker, Sir Gilbert, 80 

Wallis, B. C., 122 

Ward, R. DeCourcy, 46, 199, 200, 214 

Warmings, 49 



228 



METHODS IN CLIMATOLOGY 



Water vapor pressure, 100 

Waves, superposition of, 71 

Weather Bureau, see U. S. Weather Bureau 

Wegener, A., 178, 189 

Weighted mean, 85 

Weighted overlapping sums, 69 

Wet bulb temperature, 110 

Wet spells, 123, 124 

Wijkander, A., 187, 188 

Willett, H. C., 193, 206 

Wilson, W. M., 201 

Wind, resultant run of the, 101, 104 

resultant velocity, 101, 105 

steadiness of the, 106, 189 

"true azimuth" of the, 103 



Wind directions, frequencies of, 103 

azimuth of, 103 
Wind roses, 186 

vs. air mass analysis, 188 

for different elements, 187 
Wind "star," 186 ff. 

Wind velocity, reduction of, to the "true 
mean," 92 

resultant, 101, 105 
Winkler, M., 126, 180 
"World Weather Records," 

Yule, C. U., 162 
Zenker, W., 196